Vogt BMC Bioinformatics 2010, 11:289
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RESEARCH ARTICLE
Open Access
Spatio-structural granularity of biological material
entities
Research article
Lars Vogt
Abstract
Background: With the continuously increasing demands on knowledge- and data-management that databases have
to meet, ontologies and the theories of granularity they use become more and more important. Unfortunately,
currently used theories and schemes of granularity unnecessarily limit the performance of ontologies due to two
shortcomings: (i) they do not allow the integration of multiple granularity perspectives into one granularity framework;
(ii) they are not applicable to cumulative-constitutively organized material entities, which cover most of the biomedical
material entities.
Results: The above mentioned shortcomings are responsible for the major inconsistencies in currently used spatiostructural granularity schemes. By using the Basic Formal Ontology (BFO) as a top-level ontology and Keet's general
theory of granularity, a granularity framework is presented that is applicable to cumulative-constitutively organized
material entities. It provides a scheme for granulating complex material entities into their constitutive and regional
parts by integrating various compositional and spatial granularity perspectives. Within a scale dependent resolution
perspective, it even allows distinguishing different types of representations of the same material entity. Within other
scale dependent perspectives, which are based on specific types of measurements (e.g. weight, volume, etc.), the
possibility of organizing instances of material entities independent of their parthood relations and only according to
increasing measures is provided as well. All granularity perspectives are connected to one another through
overcrossing granularity levels, together forming an integrated whole that uses the compositional object perspective as
an integrating backbone. This granularity framework allows to consistently assign structural granularity values to all
different types of material entities.
Conclusions: The here presented framework provides a spatio-structural granularity framework for all domain
reference ontologies that model cumulative-constitutively organized material entities. With its multi-perspectives
approach it allows querying an ontology stored in a database at one's own desired different levels of detail: The
contents of a database can be organized according to diverse granularity perspectives, which in their turn provide
different views on its content (i.e. data, knowledge), each organized into different levels of detail.
1 Background
Due to the ever increasing amounts of data generated in
biomedical research, data bases for managing these data
become more important. And due to the need for more
standardization in form of formalization and externalization (see [1,2]) that accompanies this trend, bioinformatics and ontology development received more and more
attention within biomedical sciences (e.g. [3-12]).
* Correspondence: [email protected]
1
Institut für Evolutionsbiologie und Ökologie, Universität Bonn; An der
Immenburg 1, D-53121 Bonn, Germany
Full list of author information is available at the end of the article
Biology and medicine belong to the major domains of
application of ontologies. The large amounts of different
kinds of biomedical data and the organizational complexity of organisms thereby often require bio-ontologies to
provide formal tools for distinguishing data types and for
managing different levels of organizational complexity.
While ontologies accomplish the former by organizing
their terms and concepts along the lines of is-a relations
that result in an encaptic (i.e. nested) hierarchical organization of terms and concepts, they usually accomplish the
latter by differentiating levels of granularity along the
lines of part-whole relations resulting in granularity trees
(e.g. [13,14]).
© 2010 Vogt; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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When modeling their domain of reality and representing different types of biological material entities, and
their properties and relations, biologists commonly refer
to two distinct ways of hierarchically ordering them. On
the one hand, biologists use taxonomic hierarchies in
order to structure all knowledge referring to different
types of organisms, like the Linnaean zoological taxonomy of species, genera, families and orders [15], resulting
in a more or less universal referencing system of biological knowledge. On the other hand, biologists also use
partonomic hierarchies, which organize biological objects
according to their part-whole relations. These hierarchies
resemble granularity trees for biological material entities.
Several of such rather informal hierarchical systems have
been proposed in the past. Some, like Eldredge's somatic
hierarchy [16] (see also [17,18]) are focused on biological
anatomical objects, and usually include subatomic particle <atom <molecule <organelle <cell <tissue <organ
<organ system <individual organism as their typical
ranks/levels (here and in the following ' < ' is used to indicate lower-level-than relationships). Other hierarchies are
focused on typical material entities of other biological
disciplines. Ecological hierarchies (e.g. [18]), for instance,
usually include individual <population <community
<biotic province <biosphere. Some hierarchies are focused
on genetic units (genetic hierarchy; see [19]), genealogical
units (genealogical hierarchy [20]), or evolutionary units
(homology hierarchy [21]; phylogenetic trees/cladograms).
Thus, conceptualizing organisms and their parts along
the lines of part-whole relations does not represent a new
idea and has been done before in biology. It results in a
hierarchical system of anatomical structures and their
respective terms and concepts that is usually referred to
in the biological literature as levels of (biological) organization or levels of complexity [22,23] (see also scalar hierarchy [24,25]; cumulative constitutive hierarchy [19];
building block systems [26]; Theorie des Schichtenbaus der
Welt [27]). However, these rather informal approaches of
biologists are insufficient today, since they cannot be
directly utilized in databases and fail to suffice modern
standards of data management. Therefore, and considering the organizational complexity of most biological
material entities, an adequate theory of granularity is
required that is unambiguously applicable to the biomedical domain. A central question in this respect is whether
the currently applied theories of granularity really rest on
realistic assumptions about the spatio-structural organization of the material entities that belong to the biological
domain of reality, or not. Which types of hierarchical systems do biologists use, and can they be mapped onto the
commonly used granularity schemes of, for example, zoological anatomy?
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Theories of granularity are also relevant for the development of a desperately required overarching information framework [28] promoting the integration and
exchange of different types of biomedical data. Many currently highly anticipated scientific questions, like those
involved in multidisciplinary studies of gene-phene interaction or neuroscience, have to integrate very diverse
types of data and have to exchange data between different
databases and between scientists with different scientific
backgrounds. Thereby, they often have to deal with vast
amounts of various different types of data generated in
biomedical sciences today. Such integration and exchange
of data does not only involve standardization of data and
metadata [2,29,30], but requires an additional organization of data and knowledge within databases that surpasses the organization that databases usually bring
about, which is one of the key objectives of a general
granularity framework.
Current ontologies in databases use granularity frameworks that merely allow them to store knowledge and
data about their subject domain, browse and manually
annotate data, but cannot properly be used for data integration and inferencing. This is partly due to constraints
that result from a granularity framework that limits the
organization of knowledge and data to a single granularity perspective. It may be adequate for many terminologybased application ontologies [14,31], which only aim to
provide a system of terms (i.e. controlled vocabularies)
that has been built to meet very specific purposes and
needs. However, this limitation becomes very problematic for domain reference ontologies [14,31], since they
represent general-purpose resources that are developed
to support a range of very different interests and purposes. Organizing knowledge and data with such domain
reference ontologies on the basis of a single-perspective
granularity framework while satisfying these different
interests and purposes is not possible.
In the following, commonly applied granularity theories will be briefly described and subsequently exposed to
a reality check against a specific type of organization of
material entities that typically, but not exclusively, can be
found in the domain of biology. After discussing the differences between a granularity of instances (i.e. particulars; e.g. a particular individual cell) and a granularity of
types (i.e. universals; e.g. a specific type of cell) with
respect to the biological type of organization, a general
integrated spatio-structural granularity framework will
be presented and discussed that is based on Keet's general theory of granularity [32]. It accommodates different
spatial, compositional, and even scale dependent granularity perspectives that, together, enable the exhaustive
representation of all spatio-structural aspects of material
entities of the biological type of organization.
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2 Granularity and Different Types of Hierarchies
2.1 Granular Partition and Granularity Tree
In mathematics and logic a partial order is a binary relation P that is transitive (if x has relation P to y and y has
relation P to z, than x has relation P to z: (Pxy)(Pyz) T
Pxz), reflexive (x has relation P to itself: Pxx), and antisymmetric (if x has relation P to y and y has relation P to x,
than x and y are identical: ((Pxy)(Pyx) T x = y).
Granular partitions, which represent a key concept
for theories of granularity for knowledge representation
and ontology design, are based on partial ordering relations [33-36]. Granular partitions are involved in all listing, sorting, cataloging, and mapping activities and
represent hierarchical partitions consisting of cells and
subcells, with the latter being contained within the former (with the term 'cell' used here in its general, non-biological meaning). Granular partitions require a theory of
the relations between cells and subcells of a given partition that satisfies the following conditions: (i) the subcell
relation has to be transitive, reflexive, and antisymmetric
(i.e. a partial ordering relation), (ii) the existence of a root
cell (i.e. a unique maximal cell) of which all subcells are
parts, (iii) chains of nested cells are of finite length, and
(iv) if two cells overlap, then one is a subcell of the other,
which excludes partial overlap [33-36]. Additionally,
granular partitions require a theory of the relations
between cells and entities in reality (i.e. projective relation to reality [33-35]).
Granular partitions are granular insofar as they can recognize a material entity without having to recognize all its
parts (for a more general approach to granularity see 3.2
Keet's General Formal Theory of Granularity). And since
every finite partition can be represented as a rooted tree
of finite depths [33,35,37,38] (i.e. a rooted directed graph
without cycles [39]), granular partitions can be represented as granularity trees [36].
Within a granularity tree different levels of granularity
can be distinguished, with a level being a cut (sensu [40])
in the tree structure [37]. The root of a granularity tree
represents a cut itself and therewith a level of granularity,
and all immediate children of the root another cut/level.
The elements that form a level of granularity are pairwise disjoint (i.e. no entity instantiates two different types
of the same level of granularity). Levels of granularity are
exhaustive in the sense that for every entity in the domain
of interest of an ontology there exists some other entity in
the same domain which belongs to another level of granularity and the former entity stands in a partial ordering
relation to the latter, or vice versa [36]. Moreover, if the
partitioning relation is mereological (e.g., part-whole
relation), the parts exhaustively sum to the whole in a
granularity tree, i.e. all entities belonging to one level of
granularity form parts of entities of the next higher level
of granularity [36].
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Reitsma and Bittner [36] distinguish two types of granularity trees: (i) bona fide granularity trees with all entities being bona fide (i.e., entities that exist independently
of human partitioning activities - e.g. objects that possess
only physiologically discontinuous boundaries) and (ii)
fiat granularity trees with all entities being fiat (i.e.,
entities that are created by human partitioning activities e.g. entities that possess a physiologically continuous and
therefore somewhat arbitrary boundary to a neighboring
entities).
2.2 Reality Check: Cumulative Constitutive Hierarchies
By characterizing hierarchies as being based on strict partial ordering relations (transitive, antisymmetric, irreflexive (x cannot have relation P to itself: ¬Pxx); e.g. [41-43]),
some biologists [19,43,44] have differentiated four basic
types of hierarchical systems (see Fig. 1). The stronger
irreflexive binary relation of strict partial orderings can
be defined in terms of the respective weaker reflexive
relation of partial ordering (i.e. partial orderings represent the more general case [45]). A strict partial ordering
is the reflexive reduction of its corresponding partial
ordering. Therefore, the following types of hierarchies
can be discussed within the framework of partial ordering, granular partitions, and granularity trees provided
above.
1) Constitutive hierarchy [46]: "(...) the units at each
level are physical parts of the collectives at the next higher
level, where they do not have independent existences, and
this situation continues up the level of inclusiveness" ([19],
p. 25). In other words, higher level entities consist of
physically joined elements, like cells in a multicellular
organism [43,47]. Constitutive hierarchies are based on
mereological (i.e., partonomic) inclusion that results
from a proper part-whole relation (i.e. irreflexive partwhole relation) as the ordering relation. Moreover, since
all objects belonging to one level of granularity form parts
of objects of the next higher level of granularity, parts
exhaustively sum to the whole and thus represent mereological granularity trees [36] (see Fig. 1A).
2) Aggregative hierarchy [46]: "(...) the basic units are
physically independent and remain so, but are organized
into collectives that become units at the next higher level,
and these in turn are organized into collectives at a succeeding level, and so forth in ascending degrees of inclusiveness" ([19], p. 25). Put differently, higher level entities
consist of elements that are not physically connected, but
only associated with each other, like organisms in a population [43]. Depending on what one understands as 'associated', aggregative hierarchies can either be based on
mereological/meronymic inclusion that results from a
very general notion of (proper) part-whole relationships
(e.g. ecological hierarchies [18,19]), or on taxonomic
inclusion [48] that results from is-a relations (e.g. Lin-
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A)
C)
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Constitutive Hierarchy
molecule
cell
organelle
organ
Aggregative Hierarchy
B)
Cumulative Constitutive Hierarchy
D)
Cumulative Aggregative Hierarchy
orders
company
families
platoon
genera
squad
species
private
lieutenant
sergeant
captain
Figure 1 Four different Types of Hierarchies used in Biology. A) A constitutive hierarchy of molecules, organelles, cells, and organs of a multicellular
organism, in which all molecules are contained in organelles, all organelles in cells, and all cells in organs. B) The more realistic case of a cumulative
constitutive hierarchy of molecules, organelles, cells and organs of a multicellular organism, in which not all molecules are contained in organelles or
cells, and not all cells in organs. C) An aggregative hierarchy of species being aggregated to genera, genera to families and families to orders, just like
in a traditional Linnaean taxonomy. D) A cumulative aggregative hierarchy, as it can be found in the hierarchical organization of military stuff, in which
individuals with higher ranks (i.e. sergeants, lieutenants, captains) 'emerge' in aggregates of higher order (i.e. squads, platoons, companies).
naean taxonomic hierarchy;). In case of the former, they
represent mereological granularity trees (see Fig. 1C).
3-4) Cumulative hierarchy (see also somatic hierarchy
[16]): In a cumulative hierarchy, elements of levels below
the next lower level contribute to a certain level in the
hierarchy. Many multicellular organisms exhibit this type
of hierarchical organization, as for instance humans, in
which extracellularly located bone matrix and blood
plasma (both not consisting of cells) together with tissues
and organs form the whole organism [19,43,47]. In analogy to the two types of hierarchies discussed above, Valentine and May [19] distinguish cumulative aggregative
hierarchies from cumulative constitutive hierarchies.
An example for the former is the organization of military
stuff, with privates in the lowest level, squads, consisting
of privates and sergeants, in the next level, platoons, consisting of privates, sergeants, and lieutenants, in the following level, and companies, consisting of privates,
sergeants, lieutenants, and captains, in the highest level
(see Fig. 1D). An example for a cumulative constitutive
hierarchy is the organization of extracellular matrix, i.e.
ECM, in a multicellular metazoan organism. ECM is a
macromolecular formation located outside of cells. It is
not a component of cells, but it is a component of tissues
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and therefore also of organs, organ systems, and organisms (see Fig. 1B). Therefore, in most multicellular organisms, not all macromolecules are organized in cells, since
their cells are embedded in ECM. The same applies to
some fluids, which may be thought of as components of
organs but not of the tissues of these organs. In other
words, in general organs are made up of tissue, tissue of
cells, and cells of (macro-) molecules, but a (macro-) molecule is not necessarily part of a cell, a cell not necessarily
part of tissue, and tissue not necessarily part of an organ
[49].
This situation is not unique to biology. Cumulativeconstitutively organized material entities can be found in
many other domains as well. After all, not all protons and
neurons are organized within atoms and not all atoms
within molecules, which shows that the physical domain
is also organized in a cumulative constitutive way.
3 Granularity of Instances and of Types and the
Problems with Cumulative Constitutive Hierarchies
Common approaches to granularity, such as the granular
partitions and granularity tree approach discussed above,
but also Kumar et al.'s approach [13] (which will be discussed in detail further below), all have problems with
cumulative constitutively organized entities. When looking at the granularity of a particular entity and its particular parts, and thus when modeling the granularity of
instances and not of types, the granulation of cumulativeconstitutively organized entities causes only minor problems. One can, for example, partition an organ into its
direct proper parts, which are cells and the molecules
surrounding the cells (Fig. 2A). These parts all belong to
the same cut in the corresponding granularity tree and
thus to the same instance granularity level, although they
instantiate different basic types. In a next step one can
partition the cells into their organelles and their cellular
molecules surrounding these organelles. These parts all
belong to another cut in the tree and thus to the adjacently lower instance granularity level. As a last step, one
can partition the organelles into their molecules, which
provides the lowest instance granularity level (Fig. 2B).
All this can be done in line with the granular partitions
and granularity tree approach discussed above. The only
drawback with cumulative-constitutively organized entities, opposed to constitutively organized entities, is that
the mereological sum of all entities belonging to one
instance granularity level does not always sum to the
whole entity (which violates Kumar et al.'s third principle
[13]; see further below). This would only hold, if some
particular molecules would belong to more than one
level, which would, however, violate the non-partiallyoverlap principle.
Whereas modeling cumulative-constitutively organized
entities causes only minor problems when dealing with
Page 5 of 32
instances, the situation changes fundamentally when
dealing with types. Ontologies usually deal with types
and not with instances. A granularity framework for
ontologies therefore has to deal with types, and it is the
granularity of types that causes problems. Due to the
cumulative constitutive organization, different instances
of the same type can belong to different granularity levels.
In the example above, some particular molecules together
with some particular cells belong to the granularity level
directly below the 'organ' level, whereas other particular
molecules together with some particular organelles
belong to the next lower level of granularity, and still
other particular molecules belong to the lowest level of
granularity (Fig. 2B). In other words, in cumulative-constitutively organized entities the extensions of some types
(here it is 'molecule') cross the boundary between different levels of instance granularity. As a consequence, one
cannot directly transform or map the topology of an
instance granularity tree to its corresponding type granularity tree, since the resulting tree would violate granular
partitions condition (iv) (i.e. if two cells overlap, then one
is a subcell of the other, excluding partial overlap).
The question is, how one can deal with this situation in
terms of a granularity of types. The intuitive solution,
which is also reflected in the term 'cumulative constitutive hierarchy', is to rank types according to the lowest
level of granularity of their instances. This sortation-bytype functions like a granular sedimentation of all the
instances of one type to the lowest level they occupy (Fig.
2C). While this represents a pragmatic solution to the
question of how to infer type granularity, it poses significant problems for the common approaches to granularity.
All the parts of a cumulative-constitutively organized
material entity that are objects belonging to the same particular type granularity level do not exhaustively sum to
their respective wholes - not all objects belonging to one
level of type granularity form parts of the next higher
level of type granularity. This either violates one criterion
of Reitsma and Bittner's [36] mereological bona fide granularity trees, or the tree contains only two levels (i.e. an
atom-molecule level and a complex-whole level - for the
example from above, no 'organelle' and no 'cell' level). The
alternative would be that cumulative-constitutively organized material entities could only be partitioned forming
fiat granularity trees, which in case of biological anatomy
would represent a conceptualization of structural organization that contradicts all common research practice in
biology. Thus, since the part-whole relations found in
metazoan organisms are of the cumulative type [47],
cumulative constitutive organization represents the reality that all attempts of developing formally stringent representations of levels of type granularity of biomedical
material entities have to cope and to comply with. All
partitioning schemes that do not comply to this fact are
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A)
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Compositional Object Partition
a
i
f
m
k
e
j
c
B)
b
n
g
q
l
p
d
q
Cut II
m
i
e
f
e
f
n
j
cell
organelle
organ
direct_proper
_part_of
s
C)
Sortation by Type
Type Levels
Organ
s
Cut III
molecule
r
Bona Fide Instance Granularity Tree
Instance Levels
Cut IV
Cut I
h
o
a
b
c
d
a
b
c
d
a
b
c
d
Cell
r
o
k
g
h
g
h
s
p
l
Organelle
Molecule
r
q
m
i
n
e
f
j
o
a
b
c
d k
p
g
h
l
Figure 2 Granularity of Instance vs Granularity of Types. A) Compositional object partitions of an organ. Three partitions are shown: (i) into cells
(q, r) and extracellular molecules (a-d); (ii) into organelles (m-p) and cellular molecules (a-h); and (iii) into organelle molecules (a-l). B) The bona fide
granularity tree based on these three partitions. The organ itself and each of its partitions represent a cut in the tree and thus an instance granularity
level. C) The extension of 'molecule' (i.e. the distribution of instances of 'molecule') of the organ crosses the boundaries of instance granularity levels.
Therefore, the instance granularity tree cannot be directly transformed into or mapped upon a type granularity tree. But by following the simple and
intuitive rule that a type occupies the lowest granularity level of its instances, one can nevertheless infer granularity levels for types.
of limited applicability. This holds particularly for all
comparative studies that involve data from specimens of
different species or taxa.
Another problem with biological entities and their
organization within type granularity trees results from
their tremendous diversity. The claim that objects
belonging to the same level of type granularity should
have roughly the same size is not realistic. This becomes
obvious when considering the impressive variability of
real organisms, their mutability during development,
their modification of scale and proportions during
growth, their flexibility in scales and proportions under
varying environmental conditions, and their evolutionary
diversity of phenotypes across species. Already the size of
cells within one organism usually vary considerably (e.g.
oocytes, sperms, giant axons). Ear bones of some whale
species are about 100 times smaller than their ribs. Some
Caulerpa algae may grow to 3 m in length, whereas some
Mycoplasma bacteria have a diameter of only 10 μm.
els of granularity, can be found in many bio-ontologies,
like those that can be found via the BioPortal web-service
of the National Center for Biomedical Ontology [50].
Although formally more stringent than the schemes proposed by most biologists, the actual granularity schemes
of ontologies suggested so far show a considerable
amount of variety (see Table 1). This variety cannot be
explained solely by their differences in scope and domain
of application (i.e., taxonomic coverage and coverage of
granularity levels - some ontologies are restricted to a
specific taxon and/or a set of specific levels of granularity). This implies that the theories of granularity applied
(if any) obviously differ fundamentally from one another.
No commonly accepted approach to type granularity has
been agreed upon. Thus, despite its importance, how to
properly organize anatomy ontologies (particularly crossspecies anatomy ontologies) and define and demarcate
different levels of type granularity is obviously still an
open question.
4 Theories of Granularity
Schemes of type granularity, although mostly not accompanied by an explicit presentation of formally defined lev-
4.1 Kumar et al.'s Theory of Granularity and its Problems
One of the few formal theories of granularity with explicitly stated formal granularity relations and explicitly
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Table 1: Granularity Schemes used in Ontologies of Anatomy.
Levels of Granularity [ordered from fine < coarse]
Domain/
Application
Parent Class
Affiliation
Ontology/Controlled
Vocabulary
Version
'macromolecular complex' < 'organelle part' < 'organelle' < 'cell part'
< 'cell';
additional types: "envelope', 'extracellular region', 'extracellular region
part', 'membrane-enclosed lumen', 'symplast', 'synapse', 'synapse
part', 'virion', 'virion part'
genomic and
proteomic
'cellular
component'
Gene Ontology (GO)
1.689
units of structural organization: 'Biological macromolecule' < 'Cell'
< 'Portion of tissue' < 'Organ' < 'Organ system' < 'Cardinal body part'
< 'Body';
subdivisions/cardinal parts of units: 'Cardinal cell part', 'Cardinal
tissue part', 'Cardinal organ part', 'Organ system subdivision',
'Subdivision of cardinal body part';
'cross-granular' types, spanning over several levels of
granularity: 'Acellular anatomical structure', 'Anatomical cluster',
'Gestational structure', 'Vestigial embryonic structure'
human anatomy
- to a limited
degree also
vertebrate
anatomy
'Anatomical
structure'
Foundational Model
of Anatomy (FMA)
3.0
MicroscopicStructure, in no clear order: 'Cell', 'CellularStructure',
'Cilium', 'Liposome', 'MicroOrganism'
OrganicSolidStructure: 'Cell' < 'Body structure' < 'Organism'
medical coding
and
classification,
medical
terminologies,
'MicroscopicStru
cture';
'OrganicSolidStr
ucture'
Galen
1.1
'Microanatomic Structure' [i.e.: 'Macromolecular Structure' < 'Cell'] <
'Tissue' < 'Organ' < 'Organ System' < 'Body Part' - 'Body Region';
additional types: 'Body Cavity', 'Body Fluid or Substance', 'Embryologic
Structure or System',
medical coding
for clinical care,
translational
and basic
research, and
public
information and
administrative
activities
'Anatomical
Structure,
System, or
Substance'
NCI Thesaurus
09.07
No clear order: 'Body organ structure', 'Body region structure', 'Body
space structure', 'Body system structure', 'Body tissue structure', 'Body
wall structure', 'Cell structure', 'Developmental body structure', 'Entire
anatomical structure', 'Human body structure', 'Intercellular
anatomical structure', 'Non-human body structure', 'Sex structure',
'Structure of bilateral paired structures', 'Structure of multiple
topographic sites', 'Structure of product of conception', 'Transplant'
clinical terms
'Anatomical
structure'
SNOMED-CT
2009_0
1_31
'cell' < 'portion of tissue' < 'multi-tissue structure' < 'compound organ'
< 'organism subdivision' < 'whole organism'
Zebrafish
anatomy and
development
'anatomical
structure'
Zebrafish anatomy
and development
1.22
Xenopus anatomy
and development
1.19
Teleost anatomy and
development
1.120
Xenopus
anatomy and
development
'cell' < 'portion of tissue' < 'multi-tissue structure' < 'compound organ'
< 'organism subdivision' < 'body'
multispecies fish
anatomy
'anatomical
structure'
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Table 1: Granularity Schemes used in Ontologies of Anatomy. (Continued)
'Biological small molecule' < 'Biological Macromolecule' < 'Cell part' <
'Cell' < 'gross anat structure' < 'Organism';
with 'gross anat structure' comprising: 'Organ part' < 'Organ' < 'Body
part subdivision' < 'Body part' < 'Body'
vertebrate
anatomy;
derived mainly
from FMA
'Biological
object''
Basic Vertebrate
Anatomy
1.1
No clear order: 'anatomic region', 'body fluid or substance', 'organ
system'
anatomical
dictionary for
the adult mouse
'adult mouse'
Mouse adult gross
anatomy
1.195
No clear order: 'adipose tissue phenotype', 'cardiovascular system
phenotype', 'cellular phenotype', 'craniofacial phenotype', 'digestive/
alimentary phenotype', etc.
standard terms
for annotating
mammalian
phenotypic data
'mammalian
phenotype'
Mammalian
phenotype
1.298
No clear order: 'anatomical system', 'embryonic structure', 'larval
structure', 'tissue'
a structured
controlled
vocabulary of
the anatomy of
Amphibians
-
Amphibian gross
anatomy
1.8
'Anatomy', 'Cell', 'Nucleus';
with 'Anatomy' comprising: 'axis', 'body region', 'extracellular
component', 'organ', 'organism'
anatomy of C.
elegans
'C. elegans Cell
and Anatomy
Ontology'
C. elegans gross
anatomy
1.31
'cell component' < 'multi-cell-component structure' < 'cell' < 'portion
of tissue' < 'multi-tissue structure' < 'organ system subdivision' <
'organ system' < 'organism subdivision' < 'organism';
additional types: 'acellular anatomical structure', 'anatomical structure
with acellular and cellular components', developing anatomical
structure', 'endocrine organ', 'cell cluster organ', 'compound cell
cluster organ', 'sense organ'
anatomy of
Drosophila
melanogaster
'anatomical
structure'
Drosophila gross
anatomy
1.30
'cell component' < 'cells' < 'portion of tissue' < 'multi-tissue structure'
< 'compound organ' < 'organism subdivision' < 'multi-cellular
organism'
additional types: 'acellular anatomical structure', 'anatomical group',
'egg', 'extraembryonic structure',
a structured
controlled
vocabulary of
the anatomy of
mosquitoes.
'anatomical
structure'
Mosquito gross
anatomy
1.10
'acellular anatomical structure', 'portion of organism substance',
'whole organism';
remarkably with 'organ system' as a subtype of 'whole organism'
an ontology for
spider
comparative
biology
including
anatomical
parts, behavior,
and products
'anatomical
entity'
Spider Ontology
1.17
'cell component' < 'cell' < 'portion of tissue' < 'multi-tissue structure' <
'compound organ' < 'organism subdivision' < 'multi-cellular
organism'
additional types: 'acellular anatomical structure', 'anatomical group',
'bar', 'carina', 'extraembryonic structure', 'inflection', 'ruga', 'sculpture',
'sense organ', 'sulcus'
anatomy of
Hymenoptera
'anatomical
structure'
Hymenoptera
Anatomy Ontology
SVN
Revision
2804
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Table 1: Granularity Schemes used in Ontologies of Anatomy. (Continued)
'cell component' < 'cell' < 'portion of tissue' < 'multi-tissue structure' <
'compound organ' < 'organism subdivision' < 'multi-cellular
organism'
additional types: 'acellular anatomical structure', 'anatomical group',
'embryonic structure', extraembryonic structure',
anatomy of
Bilateria
'anatomical
structure'
Bilateria anatomy
-
'plant cell' < 'tissue' < 'organ' < 'whole plant'
additional types: 'gametophyte', 'in vitro cultured cell, tissue and
organ', 'sporophyte'
botanical
anatomy &
morphology
'plant structure'
Plant structure
1.64
No clear order: 'composite structure', 'fruitbody', 'hypha', 'mycelium',
'pseudohypha', 'sporophore', 'unicellular structure'
a structured
controlled
vocabulary for
the anatomy of
fungi.
'microbial
structure
ontology'
Fungal gross
anatomy
1.3
fiat object part: 'Regional Part Of Cell Component' < 'Regional Part Of
Cell'
object: 'Molecule' < 'Cellular Component' < 'Cell' - 'Extracellular
Structure'
object aggregate: 'Aggregate Object' < 'Population'
biomedical
structures from
macromolecules
to supracellular
level
'fiat object part',
'object', 'object
aggregate'
Subcellular Anatomy
Ontology (SAO)
1.2.5
'cell component' < 'cell' < 'portion of tissue' < 'multi-tissue structure' <
'compound organ' < 'organism subdivision' < 'multi-cellular
organism'
additional types: 'acellular anatomical structure', 'anatomical group',
'extraembryonic structure',
cross-species
anatomy:
platform for
integrating
different
species-specific
anatomy
ontologies
'anatomical
structure'
Common Anatomy
Reference Ontology
(CARO)
1.3
No clear order: 'acellular anatomical structure', 'anatomical group',
'anepisternum', (...), 'cell', 'cell part', 'cerebellum lobule vi', etc.
multi-species
anatomy,
generated from
union of existing
species-specific
anatomy
ontologies
'anatomical
structure'
Uber anatomy
ontology (UBERON)
1.72
No clear order: 'appendage', 'body part', 'cardiovascular system',
'craniofacial tissue', 'developmental tissue - animal', 'digestive
system', 'fat tissue', etc.
Minimal set of
terms for
anatomy
'animal
component'
Minimal anatomical
terminology (MAT)
1.1
'subatomic particle' < 'atom' < 'molecular entity' < 'poly molecular
composite Entity' [i.e. 'molecular complex' < 'structured nonbiological
entity'/'structured biological entity' [i.e. 'cellular component' < 'cell' <
'organism part' < 'organism']]
biomedicine
'material entity'
BioTop
-
Various ontologies and their partly implicit, partly explicit levels of granularity of anatomical structures (from [50]).
ranked levels of granularity that is applicable to biological
objects has been proposed by Kumar et al. [13]. By applying their theory to human anatomy, the authors distinguish 12 different levels of type granularity (i.e. Biological
macromolecule <Subcellular organelle <Collection of subcellular organelles <Cell <Collection of cells <Tissue sub-
division <Tissue <Organ part <Organ <Cardinal body
part <Organ system <Organism).
Kumar et al. [13] call objects belonging to a particular
level of granularity grains, and all the grains of a level of
granularity are marked by their own characteristic spatiostructural properties. Kumar et al. selected and demarcated different levels of granularity by paying attention to
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the different types of grains that exist in the biological
domain of reality as well as to grain-specific basic causal
principles. The reference to causal principles thereby
reflects a long tradition of theories in biology about how
to demarcate natural classes or natural kinds from artificial classes, ranging from Aristotelian essentialism to
modern concepts of natural kinds (see, e.g., [51-59]).
Kumar et al. [13] suggest seven basic principles of granularity:
1. "Each level of granularity is determined by a class or
type of grain.
2. The grains in a given level are parts of the grains in
the next higher level.
3. Every level of granularity is such that summing all
the grains together yields the entire human body.
4. The grains in a given level do not need to be all of
the same size, neither do they need to be homogeneous.
5. The grains in a given level must be smaller in size
than those entities on the next higher level of which
they are parts.
6. With each level of granularity there is associated
some specific type of causal understanding and thus
some specific family of causal laws; when one moves
up a level, then the grains on the lower levels become
causally irrelevant.
7. Some entities can change through time in such a way
that one and the same entity (an embryo, a tumour, an
organism) can occupy a sequence of different levels of
granularity in succession." ([13], p. 502).
Regarding principles 4 and 5, the authors explain that
the criterion of size for drawing dividing lines between
levels of granularity must be applied to each specific
entity or group of specific entities in succession and not
to the totality of all entities in a given level - it refers to
the actual situation found in a particular specimen and its
organization [13], and thus refers to instance granularity.
When comparing their granularity scheme for human
anatomy to their granularity principles, Kumar et al. [13]
defend their scheme since only the levels of cardinal body
parts and organ systems would violate principles 2 and 5,
because they involve grains which overlap in the sense of
sharing common parts (e.g. in humans, part of the respiratory system is in the head, another part is within the
chest).
However, the applicability of Kumar et al.'s principles of
granularity is highly questionable: Especially Kumar et
al.'s third principle is not conform with anatomic reality.
As already discussed above, in anatomy, the sum of all
grains of a given level does not always yield the entire
organism - independent of instance or type granularity.
Just think of body cavities and microflora, which are usually not included in any level of granularity [32]. Moreover, considering the genuine cumulative nature of the
Page 10 of 32
anatomical organization of multicellular metazoans, this
principle also fails the test of reality: usually, not all the
cells of a multi-cellular organism are organized within
organs. As a consequence, summing up all organs will not
yield the entire body (e.g. erythrocytes, coelomocytes,
and leukocytes are not part of any organ but part of a
human body). In this respect, the two additional and
more general principles suggested by Kumar et al. ([13],
p. 504) later on in their paper, to which they also provide
formal notations, seem to match better with biological
reality:
1. "[F] or each instance of each universal existing at a
level of granularity lower than the highest level, there
is an instance of a universal at some higher (coarser)
level of which the given instance is a part."
2. "For each instance of a universal at some level of
granularity higher than the lowest level, there is an
instance of a universal at some lower (finer) level that
is a part thereof." ([13], p. 504).
The size-principles four and five can be criticized as
well [32], since using physical size only causes problems,
such as the violation of principles already mentioned by
Kumar et al. themselves, which, according to Keet, have
to be resolved instead of handled as exceptions. Last but
not least, by limiting their theory of granularity to mereological relations only, Kumar et al.'s second principle can
be criticized for unnecessarily constraining the generality
of their theory of granularity [32].
In addition to the seven principles, Kumar et al. also
define a simple general granularity function gr, where
GRAN is the ordered set of levels of granularity (G1, G2,
G3, ..., G12) applicable to a particular domain and U is the
set of biological universals, such that gr is the function of
U onto GRAN. gr returns the level of granularity where a
particular entity resides:
gr : u → gr(u), for u ∈U and gr(u)∈ GRAN [ 13 ] .
The problem with the function is that it assumes that
the domain has been partitioned already (GRAN) and
leaves open the question of how the entities are assigned
to a level - the function requires the existence of an a priori granulated domain knowledge [32,60]. Its domaindependency limits the reusability, flexibility, and interoperability of respective computational implementations
[60]. Moreover, since gr returns only one level at a time, it
does not allow the organization of domain knowledge
according to different perspectives of granularity within
the same ontology [32], as for instance according to properties of anatomical form on the one hand, and causal dispositions (i.e. functions) or developmental properties of
biological entities on the other hand.
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4.2 Keet's General Formal Theory of Granularity
With her formal theory of granularity, Keet [32,60,61]
provides an interesting alternative that seems to circumvent some of the problems associated with Kumar et al.'s
[13] theory of granularity. According to Keet [32], granularity involves modeling something according to certain
criteria. Each model, together with its specific criteria,
forms a granular perspective. Thereby do lower levels
within a perspective contain knowledge (i.e. entities, concepts, relations, constraints) or data (i.e. measurements,
observations) that is more detailed than the next higher
level, and higher levels simplify or make indistinguishable
finer-grained details - higher levels abstract away finergrained details [61].
The most important aspect of Keet's theory is that it
allows the coexistence of different perspectives of granularity within a granularity framework. For example a
granular perspective of relative location (i.e. based on
spatial partitions) with its specific levels of granularity,
along side with a perspective of structural composition
(i.e. based on compositional partitions), one of biological
processes (i.e. based on temporal partitions), one of
causal dispositions (i.e. based on functional partitions),
and a granular perspective based on developmental relations. The possibility that different granular perspectives
can model a domain has been recognized before (e.g.,
[62]; sorts of partitions [14]; multiple valid partonomic
hierarchies [63]; perspectives &partitioning frame [64];
viewpoint dependency of hierarchy [43]). Keet [32], however, provides the first formal general theory of granularity that incorporates different perspectives within a single
domain granularity framework.
Keet [32] argues that the differentiation of different
perspectives of granularity within a domain granularity
framework implies that one can granulate a domain of
reality according to different types of granularity. Granulating reality according to different types of granularity
requires the existence of a certain type of relation that
must be specific to each particular granularity perspective. This granulation relation relates entities/types of
adjacent granular levels with one another within each
perspective. Therefore, a granularity perspective is specified and can be identified by the combination of a granulation criterion and a specific type of granularity [32].
Keet [32] not only proposes a more general approach to
granularity than the theory suggested by Kumar et al.
[13], but also provides the respective formal definitions,
axioms, and theorems accompanying her theory. Her theory is not only more general due to the fact that it allows
the combination of different perspectives of granularity
within a single framework, but also because it is not
solely based on parthood relations (i.e. mereology). It
considers taxonomic inclusion (i.e. class-subsumption
hierarchy based on is-a relations and thus on set theory)
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as another distinct but valid way of understanding granularity (see also [61]). Moreover, it accommodates both
quantitative (i.e. arbitrary scale) and qualitative (i.e. nonscale dependent) aspects of granularity.
According to Keet, a granularity framework must meet
the following criteria:
• A granularity perspective must have at least two levels of granularity.
• The entities/types in a granular level must have at
least one aspect in common, which represents the criterion of granulation by which to granulate the data,
information, or knowledge [32].
• The criterion of granulation specifies the kind/category of properties according to which the domain is
partitioned, levels identified, and the subject domain
is granulated.
• A particular granularity level may only be contained
in one perspective [65].
• Given that each granular level is contained in a granular perspective and different perspectives within a
domain granularity framework are disjoint, a particular entity/type may reside in more than one level, but
all the levels in which it is contained must belong to
distinct granular perspectives.
The advantage of Keet's approach is obvious. The
increase in generality coupled with the increase in formalism results in more flexibility and thus a broader
applicability of her theory within knowledge management systems that use ontologies. Keet's general theory of
granularity allows for a detailed and sophisticated structuring of an ontology by separating types of entities into
various different types of taxonomic and partonomic
hierarchies [32]. Like a meta-ontology, this additional
organizational layer can be formally added onto an existing ontology. The precise categorization of ontology
components by type according to, for instance, relative
location, biological processes, structural components,
causal dispositions (i.e. functions), structural and functional parthood relations, developmental properties and
relations, genealogical relations, and evolutionary origin,
facilitates the construction of a more realistic and
detailed model of biological reality.
5 General Schemes for Partitioning Complex
Structures
In the following I will distinguish three different
approaches of decomposing a complex structure solely
based on its spatio-structural properties, each of which
results in a specific type of perspective. These three basic
types of decompositions - compositional, spatial, and resolution-dependent - serve as a basis for distinguishing
three corresponding general types of type granularity
perspectives, which will be discussed in a subsequent section. Thereby, for the purpose of this paper, the Basic
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Table 2: Definitions of the Basic Formal Ontology.
Definition
Parent Class Affiliation
Link/ID
'material entity': "An independent continuant that is spatially extended whose
identity is independent of that of other entities and can be maintained through
time. Note: Material entity subsumes object, fiat object part, and object aggregate,
which assume a three level theory of granularity, which is inadequate for some
domains, such as biology.
Examples: collection of random bacteria, a chair, dorsal surface of the body"
'independent continuant'
http://www.ifomis.org/bfo/1.1/
snap#MaterialEntity
'object': "A material entity that is spatially extended, maximally self-connected
and self-contained (the parts of a substance are not separated from each other by
spatial gaps) and possesses an internal unity. The identity of substantial object
entities is independent of that of other entities and can be maintained through
time.
Examples: an organism, a heart, a chair, a lung, an apple"
'material entity'
http://www.ifomis.org/bfo/1.1/
snap#Object
'fiat object part': "A material entity that is part of an object but is not demarcated
by any physical discontinuities.
Examples: upper and lower lobes of the left lung, the dorsal and ventral surfaces of
the body, the east side of Saarbruecken, the lower right portion of a human torso"
'material entity'
http://www.ifomis.org/bfo/1.1/
snap#FiatObjectPart
'object aggregate': "A material entity that is a mereological sum of separate
object entities and possesses non-connected boundaries.
Examples: a heap of stones, a group of commuters on the subway, a collection of
random bacteria, a flock of geese, the patients in a hospital"
'material entity'
http://www.ifomis.org/bfo/1.1/
snap#ObjectAggregate
Definitions of the top-level types of material entity of the Basic Formal Ontology (BFO version 1.1).
Formal Ontology (BFO [3]) will be used as a top-level
ontology. Reference will be made to its defined type
'material entity' (i.e. a spatially extended material entity)
and its subtypes 'object' (i.e. a material entity that is maximally self-connected, possessing an internal unity), 'fiat
object part' (i.e. a material entity that is part of an object
but not demarcated by any physical discontinuity), and
'object aggregate' (i.e. a material entity that is a sum of
separate object entities, possessing non-connected
boundaries) (see Table 2 for complete definitions).
5.1 Compositional versus Spatial Partitions
While discussing different ways of partitioning anatomical structures, Mejino et al. [5] distinguish a compositional from a spatial partition (see also [11,66]). The
compositional partition is based on what they call constitutional parts, which are distinct (anatomical) objects
that are components of a structural (anatomical) whole,
and thus of an object of a higher granularity level (Fig.
3A). A compositional partition thus describes the component parts of a given object, which themselves are objects
of a lower level of granularity. In other words, whenever a
given object is partitioned and the resulting set of parts
are exclusively objects of a lower level of granularity, it
represents a compositional partition. This also holds for
the partitioning of object aggregates into their constitutional object parts.
Mejino et al.'s [5] spatial partition (see also fiat partition, [66]), on the other hand, is based on regional parts.
Regional parts are fiat object parts that result from an
arbitrary subdivision of an (anatomical) object into sets of
diverse constitutional parts that share a given location
within and relative to the object (Fig. 3B, C, and 3D). A
spatial partition thus describes the fiat object parts that
result from an arbitrary spatial division of an object.
Although arbitrary, the resulting fiat boundaries nevertheless may be determined by specific landmarks or coordinates [67], or other pragmatic or even scientifically
justified reasons, like for example differences in genetic
expression patterns of cells of different organ subdivisions. Anyhow, whenever a given object is partitioned
and the resulting set of parts consists of fiat object parts,
it represents a spatial partition. The demarcation of compositional and spatial partitions seems to be straight forward: If a partition includes only fiat object parts, it
represents a spatial partition, and if it includes only
objects, it represents a compositional partition. However,
do these two cases cover all possible spatio-structural
partitions?
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A)
Compositional Partitions
Page 13 of 32
B)
Spatial Partition
fiat organ part
fiat organ part
molecule
cell
organelle
organ
fiat boundary
C)
Spatial Partition
cell aggregate
with ECM
cell aggregate with ECM
D)
Spatial Partition
cell part aggregate
with ECM
cell aggregate with ECM
Figure 3 Examples of Compositional and Spatial Partitions. A) A combination of compositional partitions of an object into objects of a lower level
of granularity as its constitutional parts. Exemplified by three partitions of an organ: A partition into cells, one into organelles, and one into molecules.
B) A spatial partition of an object into fiat object parts as its regional parts. Exemplified by the partition of an organ into three fiat organ parts. C) A
spatial partition of an object into fiat object parts as its regional parts. Exemplified by the partition of an organ into two cell aggregates with ECM. D) A
spatial partition of an object into fiat object parts as its regional parts. Exemplified by the partition of an organ into a cell part aggregate with ECM and
a cell aggregate with ECM. ECM: extracellular matrix.
5.1.1 The Special Case of 'Object Aggregate' in Cumulative
Constitutive Hierarchies
The BFO not only distinguishes 'object' and 'fiat object
part', but also 'object aggregate'. Where do object aggregates fit into this scheme? Whereas one could argue that
for constitutive hierarchies, object aggregates may be
treated like constitutional compositions (i.e. with objects
as proper parts of object aggregates or object aggregates
as proper parts of objects that belong to the adjacent
higher granularity level), for cumulative constitutive hier-
archies this is more problematic. This is because aggregates of objects of cumulative-constitutively organized
entities are composed not only of the objects defining the
aggregate, but also of objects and fiat object parts of
lower levels of granularity. A cell aggregate of a multicellular metazoan, for instance, is not only composed of a set
of cells, but also of the molecules that form the extracellular matrix (ECM). The ECM surrounds almost all cells of
multicellular metazoans, but is not part of cells itself.
Instead, it forms a continuous matrix, and the demarca-
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tion of cell aggregates within this matrix is always accompanied by fiat boundaries through the ECM (Fig. 3C). As
a consequence, an aggregate of (bona fide) cells and a fiat
portion of ECM is not covered by (i.e. does not instantiate) BFO's concept of 'object aggregate', because the former includes fiat object parts and the latter does not.
Instead, it is instantiated by BFO's concept of 'fiat object
part'. Therefore, any partition of an entity into such fiat
aggregates is a spatial partition.
From this follows, unfortunately, that for cumulativeconstitutively organized material entities the concept of
'object aggregate' is of limited applicability. In biology, its
applicability is almost exclusively restricted to the molecular level. Although biologists usually talk about some
anatomical structures as if they are object aggregates (e.g.
cell cluster, cell aggregate), the respective structures nevertheless are fiat object parts, because they possess fiat
boundaries. Ontologies should therefore better refer to
such fiat aggregates as 'object aggregate with fiat object
part', as for example 'cell aggregate with ECM', instead of
'cell aggregate' for a cluster of cells embedded in ECM
(Fig. 3C), and they must subsume them under 'fiat object
part'.
5.1.2 Criteria for Compositional Partitions
A compositional partition can be defined by referring to a
proper parthood relation with the restriction that this
relation exists only between objects that are proper parts
of other entities (i.e. objects, object aggregates, or fiat
object parts). Moreover, the constitutive parts of a given
compositional partition may not be proper parts of one
another (i.e. they must be direct proper parts). For the
case of an organ of a multicellular metazoan, one can thus
distinguish a cellular, an organelle, and a molecular compositional partition. Thereby, due to the organ's cumulative constitutive organization, only the molecular
partition is exhaustive - the collection of all cells of a multicellular metazoan, for instance, does not sum up to the
entire organism (the ECM, e.g., is not included in this collection but nevertheless belongs to the organism).
Since any partition with objects as its parts represents a
compositional partition, depending on what is partitioned into its object components, one can distinguish
three basic types of compositional partitions: (i) a compositional object partition, in which objects are partitioned
into their constitutive object parts; (ii) a compositional
object aggregate partition (with limited applicability for
biological and all other cumulative-constitutively organized material entities), in which object aggregates are
partitioned into their constitutive object parts; and (iii) a
compositional fiat object part partition, in which fiat
object parts are partitioned into their constitutive object
parts, while ignoring the left over fiat object part components.
Page 14 of 32
Of these three basic types, the compositional object
partition takes in a special position: Since compositional
parts are objects themselves, they can also be partitioned
into their constitutive parts. This results in a hierarchical
system of compositional parthood relations, in which the
constitutive parts of one level become the objects of partition of the next lower level. Such combinations of different compositional partitions can be organized as an
encaptic hierarchy of proper parthood relations - a granular partition. This granular partition can be represented
as a tree of proper parthood relations between objects,
with each compositional object partition contributing
one level of constitutive parts, thereby forming what
Reitsma and Bittner [36] call a bona fide granularity tree.
Regarding Keet's theory, on the other hand, the combination of compositional object partitions form a compositional object granularity perspective.
5.1.3 Criteria for Spatial Partitions
The situation is different when partitioning a complex
whole into its regional parts. For defining a particular
spatial partition it is not sufficient to refer to a proper
parthood relation that has a fiat object part assigned to it,
because partitioning a given material entity into all of its
possible fiat object parts will yield many different but
ontologically still equally well justified spatial partitions.
Unfortunately, due to the arbitrariness of fiat boundaries,
these partitions are usually not congruent with one
another - "Reality is a cheese: it can be cut in many ways."
([68], p. 21). As a consequence, for a given object there
are always more spatial partitions possible than compositional partitions. Moreover, each spatial partition forms a
granularity level and not all possible levels can be combined to a single spatial granularity perspective. Put differently, there always exist several spatial granularity
perspectives that cannot be mapped onto one another.
Therefore, due to the arbitrariness of fiat boundaries,
spatial granularity perspectives always require criteria
additional to being spatial partitions.
One possibility is to link a spatial partition to a given
compositional partition. A spatial partition could thus be
defined in reference to a proper parthood relation that is
restricted to hold only between a fiat object part and its
corresponding object. However, this parthood relation
requires additional constraints: the regional parts of such
a spatial partition must represent proper parts of objects
belonging to the same compositional partition (i.e.
objects of the same compositional granularity level).
While this would still result in many different partitions,
one could differentiate between the different granularity
levels of the objects to which they refer - to each object
granularity level a set of possible spatial partitions would
be assigned.
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5.1.4 Combining Compositional and Spatial Partitions
The Foundational Model of Anatomy ontology (FMA
[11,62]) represents a well developed and frequently used
ontology for human anatomy. It has been used as a template for developing vertebrate anatomy ontologies as
well. Implicitly, it uses a type granularity framework that
combines a compositional and a spatial granularity perspective in which the spatial partition is dependent on the
compositional partition. The backbone hierarchy of the
FMA is provided by what the authors [11] call salient levels of structural organization (Biological macromolecule
<Cell <Portion of tissue <Organ <Organ system <Cardinal
body part <Body). According to Rosse and Mejino they
are defined in reference to units of structural organization
that possess only bona fide boundaries (i.e. objects and
object aggregates). Since the levels are conceived to contain only objects and object aggregates, the respective
partition would represent a compositional partition. In
addition to this compositional partition, the FMA distinguishes what the authors call transitional or intermediate
levels (Cardinal cell part <Cardinal tissue part <Cardinal
organ part <Organ system subdivision <Subdivision of
cardinal body part). They are defined in reference to subdivisions or cardinal parts of objects and thus in reference to the salient levels of structural organization. Since
these transitional levels are conceived to contain only fiat
object parts, the respective partition would represent a
spatial partition. And since they are defined in reference
to the salient levels, each spatial partition is linked to a
corresponding compositional partition, which at its turn
provides the necessary additional criterion.
Interestingly, Rosse and Mejino [62] interpret the transitional levels to represent intermediates that provide the
connection between the salient levels. Thus they seem to
implicitly combine these two granularity perspectives
within a single granularity framework. According to
Keet's general formal theory of granularity [32], such a
combination of different granularity perspectives within
the same granularity framework is possible, as long as the
perspectives are clearly separated and no granularity level
is part of more than one granularity perspective.
The granularity scheme of the FMA also bears some
problems that result from the cumulative constitutive
organization of multicellular organisms and the problems
of would-be "object aggregates" within such organizations. Most of the time they actually represent fiat object
parts and therewith regional parts rather than constitutional parts, as they are object aggregates with fiat object
parts. For instance are FMA's 'Portions of tissue', 'Organ
system', and 'Cardinal body part' not instantiated by
objects or object aggregates but actually by fiat object
parts, which do not represent constitutional parts.
Instead, they represent regional parts and can therefore
only belong to spatial and not to compositional parti-
Page 15 of 32
tions. As a consequence, FMA's salient levels of organization contain levels defined in reference to constitutive
parts alongside with levels defined in reference to
regional parts. Such a mixture of compositional and spatial partitions within one granularity perspective, however, is formally inconsistent and should be avoided.
Since FMA's transitional levels are defined in reference
to their respective salient levels, this problem also affects
the transitional levels: 'Cardinal tissue part', 'Organ system subdivision', and 'Subdivision of cardinal body part'
are instantiated by regional parts of fiat object parts and
therewith by regional parts of regional parts. As such,
they are lacking the necessary additional criterion, since
they are not defined in reference to a respective granularity level of a compositional perspective.
A similar problem applies to Kumar et al.'s [13] granularity scheme for human anatomy. It is responsible for
their levels of 'cardinal body part' and 'organ system' failing to meet their granularity principles two (i.e. grains in
a given level are parts of grains in the next higher level)
and five (i.e. grains in a given level must be smaller in size
than those entities on the next higher level of which they
are parts). Part of the human respiratory system is
located in the head and another part is within the chest,
because cardinal body parts and organ systems obviously
belong to two distinct and incongruent regional partitions and thus to two different spatial granularity perspectives. This becomes also apparent when considering
the granular partitions criterion of non-partial overlap
mentioned earlier (see 2.1 Granular Partition and Granularity Tree), since it implies that the partition of human
anatomy into organ systems and into cardinal body parts
together do not form a granular partition and thus also
no granularity tree. One has to keep in mind that levels
belonging to different perspectives should never be combined within a single synthetic perspective but rather
within a general granularity framework.
5.2 Scale-Based versus Qualitative Partitions
A notion commonly associated with granularity is that of
size. For example, it is often claimed that entities belonging to the same level of granularity should have roughly
the same size. While such requirements have been identified to be problematic (see, e.g., Kumar et al.'s fourth
principle: grains in a given level do not need to be all of the
same size [13]), size relations still seem to be important to
theories of granularity (Kumar et al.'s fifth principle:
grains in a given level must be smaller in size than those
entities on the next higher level of which they are parts
[13]). However, one has to be very careful when requiring
specific size relations in connection with granularity
schemes. The actual size values may vary significantly not
only among the particular instances of a given type but
also among different moments in time for the same individual entity. Therefore, size relations should always refer
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to actual conditions and the organization found within
one and the same particular entity at a specific moment
in time and should not be claimed to exist time-independently and between parts belonging to different particular entities [13].
Rector et al. [69] distinguish qualitative from quantitative granularity and recommend to use the term collectivity for qualitative granularity and the term size range for
scale-based granularity. According to Rector et al. [69], a
collective represents an emergent whole which consists of
grains that represent the granular parts of the collective.
In other words, entities considered individually at one
level of granularity are considered as collectives with
emergent properties at the next higher level, and between
entities of lower and higher levels exist part-whole relations. The two types of granularity perspectives discussed
above, based on regional and constitutive parthood relations, represent examples for qualitative granularity.
According to Rector et al. [69], scale-based size range, on
the other hand, is not based on a parthood relation
between grains and collectives but on a relation of large
and small: grains cannot be physically larger than their
collectives.
But then, Keet [32] distinguishes between scale dependent and non-scale dependent granularity perspectives,
which she treats as fundamentally different types of granularity perspectives within her formal theory of granularity. As she notes, "[f ] or scale-dependency it is important
to recognise that it is the representation of the entity (/
type) that counts. The representation of the real-world
entity (/type) is different at different levels of granularity
and the representation is the resultant of the combination
of the entity (/type) and the scale at which it is considered
(...)" (emphasis taken from the original; [32], p. 84).
According to her taxonomy of types of granularity, Keet
[32,65] distinguishes between grainsize (i.e. scale on
entity), with the subtypes resolution (e.g. cell wall as line,
lipid bi-layer, or 3D structure) and size of the entity (e.g.
coin separator of a vending machine) on the one hand,
and aggregation (i.e. scale of entity), with the subtypes
overlay aggregated (e.g. map of earth with more/less isotherms) and entities aggregated according to scale (e.g.
second, minute, hour) on the other hand, as two basic
types of scale dependent granularity. The criterion of
granulation of scale dependent granularity perspectives is
a combination of a quality property and a sortal property,
as for instance the surface of a structure and a surface
metric scale. The relation between entities belonging to
adjacent levels of scale dependent granularity thereby
maps onto a parthood relation [32] that combines proper
parthood with a math function for conversion of units of
measurement.
Page 16 of 32
5.2.1 Why We Need Scale Dependent Granularity
The importance of scale dependent granularity perspectives results from pragmatic considerations regarding the
overwhelming richness in detail that is inherent to nature
and the fundamental cognitive and general epistemological constraints that every particular scientist and every
actual scientific enterprise has to deal with. For example,
if one wants to investigate the social behavior of a specific
individual animal, as a first step it would usually not be
very efficient to analyze its total structural composition
on a molecular level. This would not only consume too
much time and money (two very valuable resources), but
would produce also significant amounts of irrelevant
data. Thus, depending on the purpose of an investigation,
one usually ignores specific levels of detail (i.e. levels of
granularity) and focuses on others. If a morphologist
studies the anatomy of a multicellular metazoan specimen using gross anatomy preparation methods, she usually cannot make any statements about the number and
spatial arrangements of individual cells. At this (granular)
level of resolution, she can only speak of cell aggregates as
certain portions of tissue, which represent certain
amounts or portions of matter.
Portions of matter do not represent countable units (i.e.
count-nouns), but require some arbitrary, although standardizable, quantification measure, like a glass of water
or 1 m3 of soil, therewith representing what is generally
referred to as mass-nouns. The distinction of mass-noun
and count-noun, however, is not absolute, but a result of
granular focus (resolution). A glass of water, for instance,
which is represented by a mass-noun, can be decomposed
on the molecular level into an aggregate of a specific
number of water molecules, which at its turn is a representation that uses a count-noun. Whether a count-noun
or a mass-noun is used depends largely on the individual
interests and purposes of the investigator: if one is interested in the molecular composition of a particular wine,
one would treat it as a discrete aggregate of molecules,
whereas if one is only interested in consuming a certain
amount of it, one treats it as a continuous aggregate that
has to be partitioned according to volume measures. The
same applies when investigating the anatomy of multicellular metazoans: what represents a portion of tissue at a
coarser level of resolution/granularity can be decomposed on a finer grained level into a cell aggregate with a
portion of molecular substance: a specific number of individual cells accompanied by a portion of ECM. Therefore,
the distinction between portion of tissue and cell aggregate with portion of ECM is not a matter of qualitative
properties but only a matter of resolution and, thus, of
scale (Fig. 4).
Obviously, contrary to non-scale dependent granularity, scale dependent granularity is not always exclusively
based on properties of material entities that exist inde-
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A)
Molecular Level
Page 17 of 32
B)
Organelle Level
portions of molecular substance
C)
Cellular Level
portion of ECM
cell cluster with
portion of ECM
D)
portion of organelles
and cytoplasm
Organ Level
portion of tissue
molecule
cell
organelle
organ
fiat boundary
Figure 4 Scale dependent Resolution. A) On the molecular level, individual molecules and aggregates thereof can be distinguished and are referred
to by using count-nouns (i.e. 'molecule'; 'molecule aggregate'). B) On the organelle level, individual organelles can be distinguished and are referred
to by using count-nouns (i.e. 'organelle'). Individual molecules and aggregates thereof, however, are usually not distinguished anymore and one refers
to them by using mass-nouns, such as 'portion of molecular substance'. One can distinguish extra-organelle from intra-organelle portions of molecular
substance. C) On the cellular level, individual cells can be distinguished and are referred to by the count-nouns 'cell' or 'cell cluster with ECM' (or its
resolution-dependent countable representation 'cell cluster with portion of ECM'). Individual organelles, however, are usually not distinguished anymore. Instead, one refers to the extracellular matter as non-countable 'portion of ECM', whereas the intracellular matter represents a granular mixture
of non-countable organelles and cytoplasm and is referred to as 'portion of organelles and cytoplasm'. Both represent mass-nouns. D) On the organ
level, individual cells are usually not distinguished anymore. Instead, one refers to the matter contained in organs as a 'portion of tissue', which is the
resolution-dependent non-countable representation of 'cell aggregate with ECM'. 'Portion of tissue' therewith represents a mass-noun. ECM: extracellular matrix.
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pendently of human activities. Some types of scale
dependent granularity, as for example resolution granularity, are based on properties and relations resulting
from the interaction of a human being with the material
objects around her (including other human beings and
herself ). This involves epistemological and cognitive constraints that affect our representations of real entities.
Whereas this might tell us more about our cognitive system than about the entities themselves, it is nevertheless
real and we have to deal with it when doing science.
Therefore, it is advisable to carefully distinguish such
epistemological types of scale dependent granularity,
which depend on peculiarities of human cognition (e.g.,
resolution), from ontological types of scale dependent
granularity (e.g., size of the entity). Anyhow, one must
realize that although quantitative scale dependent and
qualitative non-scale dependent types of granularity represent effectively independent types of granularity perspectives [69], and epistemological and ontological types
of scale dependent granularity must be distinguished
carefully. All of them are essential and required in scientific research practice.
5.2.2 'Object Cluster' versus 'Portion of Substance'
Considering resolution and the resulting granularity relation between count-nouns and mass-nouns, and considering the specific conditions found in cumulative
constitutive organizations, it follows that all entities that
are object aggregates or that are fiat object parts that possess aggregates of objects as their parts (e.g. cell aggregates with ECM) can be represented in two different
ways.
The example from above with a countable and a noncountable representation of cell aggregate with ECM (Fig.
4) shows: While in reality the entities of a particular type
of 'cell aggregate with ECM' that are associated with different resolution levels are the same, their representation changes according to resolution. A cell aggregate
with ECM can be represented with two different types of
resolution-dependent concepts - two mappings from the
same real world entity. Put differently, it instantiates two
different types of representation (see [32]) - a type for
which the amount of cells in the aggregate is countable
and a type for which it is non-countable.
This holds not only for fiat object parts that possess
object aggregates as their parts, but also for object aggregates. Both require at least two distinct representations.
In case the objects in the aggregate are non-countable,
the term 'portion of substance' should be used for the
coarser grained representation. In case the objects in the
aggregate are countable and one refers to them as a
countable cluster of objects, the term 'object cluster'
should be used for object aggregates and 'object cluster
with portion of substance' for object aggregates with fiat
object parts for the finer grained representation. Thus,
Page 18 of 32
for instance an aggregate of six cells embedded in ECM
should be referred to as a 'cell cluster with portion of
ECM' if one refers to the cells as individual objects. However, if the presence of the individual cells is irrelevant,
one should use the term 'portion of tissue', which represents the coarser grained representation. Anyhow, since
both 'cell cluster with portion of ECM' and 'portion of tissue' are alternative types of representation of a cell aggregate with ECM, one could also refer to the latter if one
does not want to account for resolution. This applies to
object aggregates accordingly.
6 A General Integrated Spatio-Structural
Granularity Framework for CumulativeConstitutively Organized Material Entities
Cumulative-constitutively organized material entities
usually possess a complex structure. Modeling their
structural properties more or less comprehensively
requires a general theory of granularity that allows to
combine several different type granularity perspectives
within one integrated type granularity framework. In
order to develop such an integrated type granularity
framework one first has to identify all relevant basic types
of material entity and then analyze all possible qualitative
and scale dependent relations among them that are relevant for modeling their spatial organization - their form
or morphology. In other words, all relevant major compositional and spatial partitions have to be identified and
formally covered, and where appropriate, modeled as
separate type granularity perspectives.
6.1 Why Not Kumar et al.'s Theory of Granularity
The granularity theory of Kumar et al. [13] does not meet
the criteria required for such a task. Their second principle of granularity (i.e. grains in a given level are parts of
the grains in the next higher level) constrains granularity
on qualitative parthood relations (i.e. mereology-only
relations) and thus excludes the possibility to model scale
dependent granularity such as resolution. Resolution,
however, is a key concept in most empirical experimental
sciences and should be accounted for by any model of
reality. Kumar et al.'s third principle (i.e. every level of
granularity is such that summing all the grains together
yields the entire human body), although a common principle in mereological theories, would prohibit for
instance a granularity level 'cell' for human anatomy, since
human bodies represent cumulative-constitutively organized material entities. The sum of all cells does not yield
the corresponding whole human body. This is due to the
fact that, besides various types of extracellular fluids and
different kinds of body cavities, a very important part of
the human body consists of extracellular matrix (ECM),
which holds the cells together. ECM, extracellular fluids,
and body cavities, however, are not covered by an exclu-
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sively cellular model of human anatomy. Despite this
inconsistency, Kumar et al. [13] themselves suggest a 'cell'
granularity level and do not even discuss the resulting
problems regarding their third principle. This can be
understood as an indication to their third principle not
being necessary. It unnecessarily restricts their granularity theory (this criticism also applies to the theory of
granular partitions and granularity trees when applied to
types and not to instances; see Chapter 2.1 and discussion
in Chapter 7).
Since Kumar et al.'s theory of granularity is narrow in
scope, constrained on parthood relations only, and,
moreover, restricted to accommodate only a single granularity perspective, it is not adequate for modeling spatiostructural granularity of cumulative-constitutively organized material entities. The general theory of granularity
suggested by Keet [32], on the other hand, is less restrictive and can even accommodate multiple granularity perspectives, integrated into a common granularity
framework. Her theory thus provides a theoretical framework that perfectly suits the requirements of modeling
spatio-structural granularity of cumulative-constitutively
organized material entities.
6.2 Identifying and Demarcating the Different Types of
Structural Granularity Perspectives
6.2.1 The Subject Domain
The here proposed general integrated structural granularity framework is based on Keet's general theory of
granularity [32]. Its subject domain is restricted to material entities that exhibit a cumulative constitutive organization. Taking the Basic Formal Ontology (BFO [3]) as a
top-level ontology, its basic types of 'material entity' 'object aggregate' (although of limited applicability),
'object', and 'fiat object part' - represent foundational
types within the here proposed granularity framework.
qualitative
Compositional
Object
Granularity
G : Granularity Value
'Organelle'
With them, all possible types of material entity that possess structural integrity (i.e. no part separated from all
other parts by a gap) can be classified. Every possible
arbitrary fiat part as well as every possible bona fide part
of a cumulative-constitutively organized material entity is
covered in principle.
6.2.2 The Backbone: A Compositional Object Granularity
Perspective
The three types of partitions discussed in the previous
section (5 General Schemes for Partitioning Complex
Structures) - two qualitative types of partitions (i.e., compositional and spatial) and one quantitative type (i.e. resolution), provide the core for the different types of
granularity perspectives presented here.
The most important granularity perspective of the
framework is based on compositional partition. It is
important because it provides a reference backbone to
the other perspectives (here, and in the following, if not
stated otherwise, granularity refers to type granularity).
This compositional object granularity perspective
(short: CO perspective) is based on a direct proper parthood relation between different subtypes of 'object' (see
Fig. 5). Thus, its granulation criterion combines the
defining properties of 'object' with the direct proper (constitutive) parthood property:
'object' is-direct-proper-part-of 'object';
'object' has-direct-proper-part 'object'.
As a consequence, all objects that possess a direct
proper parthood relation to at least one other object
belong to this perspective.
Following Keet, the CO perspective thus has granulation of the non-scale dependent single-relation-type (nrG
[32]) granularity type, also called the non-scale dependent primitive (npG [61]) granularity type. It uses the
direct proper parthood relation as its granulation relation.
'Cell'
'Organ'
direct
proper
part of
G=X-1
direct
proper
part of
G=X
G=X+1
Figure 5 Compositional Object Granularity Perspective. A particular qualitative compositional object granularity perspective that is defined in reference to a combination of direct proper parthood and the defining properties of 'object' as its granulation criterion. The levels of granularity are demarcated according to the defining properties of different subtypes of 'object' that possess proper parthood relations to other such subtypes (e.g.
'Organelle', 'Cell', 'Organ'), and they are ordered according to the direction of direct-proper-part-of relations.
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The different granularity levels of a CO perspective are
defined and demarcated from one another in reference to
the defining properties of those subtypes of 'object' of a
given ontology that possess proper parthood relations to
other subtypes of 'object'. As a consequence, each granularity level has at least one particular subtype of 'object'
assigned to it. However, in order to constitute a granularity perspective, there have to be at least two levels of
granularity and thus two distinct and disjoint subtypes of
'object' in a given ontology [32].
The lowest level of granularity refers to the particular
'object' subtype of which at least one instance is a direct
proper part of an instance of another 'object' subtype and
of which no instance has an instance of another subtype
of 'object' as its proper part. This way, entities residing in
adjacent granularity levels are related to one another
according to the direct proper parthood granulation relation. In other words, the granularity levels are demarcated from one another according to the properties of the
subtypes of 'object' of a given ontology that possess
proper parthood relations and they are ordered according
to the direct proper parthood granulation relation.
For instances of cumulative-constitutively organized
material entities holds:
1. An object entity is not necessarily a proper part of
some object entity that belongs to the adjacent higher
level of CO granularity.
2. Every object entity that does not belong to the lowest level of CO granularity has at least two object entities as its proper parts.
3. The object entity that is granulated represents the
maximum object entity belonging to the highest CO
granularity level. Every other object entity belonging
to this granulation is a proper part of this maximum
object entity. Thus, put differently, all object entities
that belong to a granulation are proper parts of one
maximum object entity. However, due to the cumulative constitutive organization, the object entities that
are direct proper parts of the maximum object entity
not necessarily belong to the second highest level of
CO granularity but may belong to any level of the
granulation.
6.2.3 Five Additional Compositional Granularity Perspectives
Additionally to the compositional object granularity perspective one can differentiate five other types of compositional granularity perspectives. A compositional object
of fiat object part granularity perspective that is based
on a direct proper parthood relation between a specific
subtype of 'fiat object part' and its corresponding parts
that represent the matching subtypes of 'object' (for
examples see Fig. 6). A compositional object aggregate
of fiat object part granularity perspective, in which the
corresponding parts are the matching subtypes of 'object
aggregate' based on a proper parthood relation. A compo-
Page 20 of 32
sitional object of object aggregate granularity perspective, which is based on a direct proper parthood relation
between a specific subtype of 'object aggregate' and its
corresponding subtype of 'object' as its part. And a compositional object aggregate of object granularity perspective, which is based on a proper parthood relation
between a specific subtype of 'object' and its corresponding subtype of 'object aggregate' as its part. Furthermore,
a compositional object aggregate of object aggregate
granularity perspective, which is based on a proper
parthood relation between specific subtypes of 'object
aggregate'. In case of cumulative-constitutively organized
entities, however, due to the limited applicability of the
type 'object aggregate', all additional compositional granularity perspectives are of limited relevance, except for
the compositional object of fiat object part granularity
perspective.
The granulation criterion of each additional compositional perspective combines the (direct) proper parthood
property with the defining properties of either a specific
subtype of 'object' that is assigned to a particular granularity level of the CO perspective or of the specific corresponding subtype of 'object aggregate' or 'fiat object part'
respectively:
i) 'object' is-direct-proper-part-of 'fiat object part';
'fiat object part' has-direct-proper-part 'object'.
And of limited relevance:
ii) 'object aggregate' is-proper-part-of 'fiat object
part';
'fiat object part' has -proper-part 'object aggregate';
iii) 'object' is-direct-proper-part-of 'object
aggregate';
'object aggregate' has-direct-proper-part 'object';
iv) 'object aggregate' is -proper-part-of 'object';
'object' has -proper-part 'object aggregate';
v) 'object aggregate' is -proper-part-of 'object aggregate';
'object aggregate' has -proper-part 'object aggregate'.
As a consequence, only objects that belong to the same
CO granularity level and their corresponding object
aggregate and fiat object part counterparts belong to the
same particular compositional perspective. From this follows that for object aggregates and fiat object parts there
exist as many different particular additional compositional perspectives as there are different levels of CO
granularity (except for the maximum object and thus the
highest level of CO granularity, as its instances by definition and de facto cannot be part of any higher level
object). The additional compositional perspectives are
demarcated from one another according to the defining
properties of the respective 'object' or 'object aggregate'
subtypes that must belong to the same CO granularity
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Page 21 of 32
qualitative
Compositional
Cell of Fiat
Organ Part
Granularity
X<G<X+1
'Fiat
organ part'
G : Granularity Value
'Cell aggregate
with ECM'
qualitative
Compositional
Cell of Cell Aggregate
with ECM
Granularity
'Cell cluster
with portion
of ECM'
'Portion
of tissue'
direct
proper
part of
'Fiat organ part
gg g
aggregate'
'Cell'
direc
direct
prop
p
p
proper
part of
X<G<X+1
direct
proper
part of
G=X
qualitative
Compositional
Cell of Fiat Organ
Part Aggregate
Granularity
X<G<X+1
Figure 6 Additional Compositional Granularity Perspectives. A combination of three particular qualitative compositional object of fiat object part
granularity perspectives. Each perspective is defined in reference to a combination of direct proper parthood and either the defining properties of specific subtypes of 'object' (in the perspectives depicted it is 'cell'), all of which belong to the same compositional object granularity level, or the defining
properties of their respective type of 'fiat object part', of which the object entities are direct proper part of (in the perspectives depicted these are 'cell
aggregate with ECM', 'fiat organ part', and 'fiat organ part aggregate'). Each of these additional compositional granularity perspectives contains only
two levels: a lower level, to which subtypes of 'object' belong (e.g. 'cell'), and a higher level, to which subtypes of 'fiat object part' belong (e.g., 'cell
aggregate with ECM', 'fiat organ part', or 'fiat organ part aggregate'). Notice that the three perspectives shown overlap with one another at their 'cell'
levels - a particular cell can belong to all three perspectives at the same time. ECM: extracellular matrix.
level. In other words, each particular compositional perspective is either directly or indirectly assigned to a particular CO granularity level.
Each of these particular compositional perspectives has
granulation of the non-scale dependent single-relationtype granularity type (nrG [32]). They use the (direct)
proper parthood relation as their granulation relation
and are in this respect compatible to the CO perspective.
Contrary to the CO perspective, however, each of these
additional compositional perspectives consists of only
two granularity levels (except for the compositional
object aggregate of object aggregate perspective, which
can have more than two levels). The granularity level to
which the parts belong represents the lower level, and
entities belonging to this lower level are related to entities
belonging to the higher level through (direct-) properpart-of relations.
By distinguishing different subtypes of 'fiat object part'
one can differentiate further subtypes of compositional
granularity perspectives, which, however, would go
beyond the scope of this paper (for examples see Fig. 6, 7,
and 8).
For instances of cumulative-constitutively organized
material entities holds:
1. Every object aggregate entity must have some
object entity as its proper part that belongs to the
same CO granularity level as its name giving object.
2. Not every object entity that belongs to the same
CO granularity level as the name giving object of an
object aggregate or a fiat object part must be proper
part of a corresponding object aggregate or fiat object
part entity.
3. Every fiat object part entity must have some object
entity as its proper part that belongs to a lower CO
granularity level than its name giving object.
6.2.4 Two Basic Spatial Granularity Perspectives
With the CO perspective and its different granularity levels in the background, one can differentiate two different
types of basic spatial granularity perspectives. A spatial
fiat object part of object granularity perspective, which
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General
Integrated
SpatioStructural
Granularity
Framework
Page 22 of 32
qualitative
qualitative
Compositional
Cell of Cell
Aggregate
with ECM
Granularity
Spatial
Cell Aggregate
with ECM
of Organ
Granularity
'Cell aggregate
with ECM'
'Cell aggregate
with ECM'
'Cell c
cluster
portion
with p
of E
ECM'
'
'Portion
of tissue'
'Cell cluster
with portion
of ECM'
'Portion
of tissue'
scale dependent
Resolution
of Cell Aggregate
with ECM
Granularity
proper
countable
part of
X<G<G<X+1
cl.
direct
proper
part of
Backbone
Perspective
qualitative
'Organelle'
X<G<X+1
tiss.
cl.
X<G<X+1
'Cell'
G=X-1
cl.
tiss.
proper
part of
X<G<X+1
cl.
X<G<X+1
'Cell part
aggregate
with ECM'
'Organ'
direct
proper
part of
Compositional
Object
Granularity
X<G<G<X+1
direct
proper
part of
proper
part of
G=X
G=X+1
qualitative
Spatial
Cell Part Aggregate
with ECM of Organ
Granularity
X<G<X+1
proper
part of
X<G<X+1
G : Granularity Value
'Fiat
organ part'
Overcrossing
Ov
O
veerrcr
ver
cros
ossing
sing
si
ng granularity
grra
an
anu
nu
ula
lari
rity
ty
ty
perspectives:
pers
pe
rspe
spe
p ccttiivves
es:
qualitative
Granularity
Gra
G
Gr
ranul
ra
ullari
arity llev
levels
eevels
els
off whi
which
w
wh
hich
ch the
tth
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conte
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nt
overlaps
between
ovve
ove
verla
rrla
lap
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pss b
be
bet
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ween
we
dif
di
iffffer
feerent
e gr
en
ent
ggranularity
ranu
anu
ularrity
ular
itty
different
per
p
er
erspe
specti
ctives
ct
tive
ves
eess
perspectives
Spatial
Fiat Organ Part
of Organ
Granularity
Figure 7 The General Integrated Spatio-Structural Granularity Framework. Some of its granularity perspectives are shown and how they relate,
i.e. overcross, with one another.
is based on a proper parthood relation between a specific
subtype of 'object' and its corresponding parts, which at
their turn must represent regional parts and thus specific
subtypes of 'fiat object part' (for examples see Fig. 9). Furthermore, although of less relevance for cumulative-constitutively organized entities, a spatial fiat object part of
object aggregate granularity perspective, which is
based on a proper parthood relation between a specific
subtype of 'object aggregate' and specific subtypes of 'fiat
object part' as its corresponding parts.
The granulation criterion of these two different types of
particular basic spatial granularity perspectives thus
combines the proper parthood property with the defining
properties of either a specific subtype of 'object' or 'object
aggregate' that is directly or indirectly assigned to a particular granularity level of the CO perspective or of the
specific corresponding subtype of 'fiat object part'.
i) 'fiat object part' is-direct-proper-part-of 'object';
'object' has-direct-proper-part 'fiat object part'.
And of limited relevance:
ii) 'fiat object part' is-direct-proper-part-of 'object
aggregate';
'object
aggregate' has-direct-proper-part 'fiat
object part'.
As a consequence, only objects that belong to the same
CO granularity level, or their respective object aggregates, and their corresponding fiat object part parts
belong to the same particular spatial granularity perspective. From this follows that for each type of basic spatial
granularity there exist as many different particular perspectives as there are different levels of CO granularity.
The particular spatial perspectives are demarcated from
one another according to the defining properties of the
respective 'object' or 'object aggregate' subtypes, which at
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Page 23 of 32
Parthood
Pa
n
Levels
Parthood
Pa
arthood
two
Levels
Parthood
n
Levels
object
Pa
Parthood
arthood
2 Levels
Parthood
d
Levels
2 Leve
els
object aggregate
is of limited
relevance for
cumulative-constitutively
organized entities
'Cell'
'Organ'
Parthood
2 Levels
Parthood
2 Levels
fiat object part
Parthood
rthood
Levels
Partho
Parthood
'Cell part
aggregate
with ECM'
M
'Organ part
aggregate with
portion
of tissue' Levels
porrt
all
Parthood
2 Levels
Parthood
Partho
ood
Levels
2 Le
Lev
evels
Parthood
Pa
Par
arrthood
thood
2 Levels
Partho
Parthood
Parthood
many
Levels
Compositional Granularity Perspective
Spatial Granularity Perspective
'Cell aggregate
with ECM'
two
Levels
'Fiat
Organ part'
Levels
Parthood
2 Levels
Parthood
Resolution Granularity Perspective
Parthood
2 Levels
Parthood
2 Levels
Parthood
2 Levels
Figure 8 Overview of the Framework. Overview of the different types of compositional, spatial, and resolution granularity perspectives of the General Integrated Spatio-Structural Granularity Framework and how they relate to the basic subtypes of 'material entity', with 'Cell', 'Organ', 'Cell aggregate with ECM', 'Cell part aggregate with ECM', 'Organ part aggregate with portion of tissue', and 'Fiat organ part' as examples. The arrows indicate
whether it is a compositional, a spatial, or a resolution perspective and how many granularity levels it distinguishes. Different subtypes of 'fiat object
part' are distinguished (i.e. 'Cell part aggregate with ECM', 'Organ part aggregate with portion of tissue', 'Cell aggregate with ECM', and 'Fiat organ
part'), which allow the differentiation of further subtypes of spatial and compositional granularity perspectives, which, however, are not discussed in
detail here. All perspectives involving object aggregates are not covered in detail, since they are of limited relevance for cumulative-constitutively
organized entities. ECM: extracellular matrix.
their turn must belong either directly or indirectly to the
same CO granularity level. Put differently, each particular
basic spatial granularity perspective is assigned to a particular CO granularity level.
Each particular basic spatial granularity perspective has
granulation of the non-scale dependent single-relationtype granularity type (nrG [32]).
Each consists of only two granularity levels, which are
defined and demarcated according to the properties of
the 'object' or 'object aggregate' subtypes versus those of
the 'fiat object part' subtypes that the perspective contains. The granularity level to which the 'fiat object part'
subtypes belong represents the lower level, and entities
belonging to this lower level are related to entities
belonging to the higher level through proper-part-of relations.
Therefore, despite the arbitrariness that is connected to
all partitions involving regional parts, these parts all have
a proper parthood relation to their respective object or
object aggregate entity in common. As a consequence, no
matter how one actually partitions a given object or
object aggregate entity into regional parts (e.g., see Fig. 9),
these parts will always represent proper parts of the given
object or object aggregate entity. They will always belong
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Page 24 of 32
qualitative
Spatial
Cell Aggregate
with ECM
of Organ
Granularity
'Cell aggregate
with ECM'
'Cell cluster
with portion
of ECM'
'Portion
of tissue'
X<G<X+1
G : Granularity Value
qualitative
Spatial Fiat
Organ Part of Organ
proper
part of
'Fiat
organ part'
'Cell part aggregate
with ECM'
'Organ'
proper
part of
qualitative
Spatial
Cell Part Aggregate
proper
part of
with ECM of Organ
Granularity
Granularity
X<G<X+1
G=X+1
X<G<X+1
Figure 9 Basic Spatial Granularity Perspectives. A combination of three particular qualitative spatial fiat object part of object granularity perspectives.
Each perspective is defined in reference to a combination of proper parthood and either the defining properties of specific subtypes of 'object' (in the
perspectives depicted it is 'organ'), all of which belong to the same compositional object granularity level, or the defining properties of their respective
'fiat object part' parts (in the perspectives depicted these are 'cell aggregate with ECM', 'fiat organ part', and 'cell part aggregate with ECM'). Each basic
spatial granularity perspective contains only two levels: a lower level, to which subtypes of 'fiat object part' belong (e.g., 'cell aggregate with ECM', 'fiat
organ part', or 'cell part aggregate with ECM'), and a higher level, to which subtypes of 'object' belong (e.g., 'organ'). Notice that the three perspectives
shown overlap with one another at their 'organ' levels - a particular organ can belong to all three perspectives at the same time. ECM: extracellular
matrix.
to the lower granularity level of its respective basic spatial
perspective and the corresponding object or object aggregate entity to the higher one.
By distinguishing different subtypes of 'fiat object part'
one can differentiate further subtypes of spatial granularity perspectives, which, however, would go beyond the
scope of this paper (for examples see Fig. 7, 8, and 9).
For instances of cumulative-constitutively organized
material entities holds:
1. Not every fiat object part entity must be proper
part of an object or object aggregate entity that is
directly or indirectly assigned to the adjacent higher
CO granularity level (e.g., not every portion of tissue
is proper part of some organ).
2. Every fiat object part entity must be proper part of
some object entity that belongs to the same CO granularity level as its name giving object entity.
3. Every object or object aggregate has at least two fiat
object part entities as its proper part.
4. For a given granulation, every fiat object part entity
must be proper part of the same maximum object
entity.
6.2.5 Two Basic Scale Dependent Resolution Granularity
Perspectives
Besides the various qualitative granularity perspectives
presented above one can differentiate several quantitative
(i.e. scale dependent) structural granularity perspectives
in reference to different subtypes of 'object aggregate' or a
specific type of fiat object part, 'object aggregate with fiat
object part'. A resolution of object aggregate granularity perspective, which is of limited relevance for cumulative-constitutively organized entities, and a resolution of
object aggregate with fiat object part granularity perspective. Each type of resolution granularity perspective
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Page 25 of 32
'Fiat object
part'
is_a
'Cell aggregate
with ECM'
is_representation_of
scale dependent
'Cell cluster
with portion
of ECM'
is_representation_of
'Portion
of tissue'
Resolution
of Cell Aggregate
with ECM
Granularity
proper
countable
part of
non-countable
representation
countable
representation
G : Granularity Value
X<G<X+1
cl.
X<G<G<X+1
cl.
tiss.
Figure 10 Resolution Granularity Perspective. A particular quantitative scale dependent resolution of object aggregate with fiat object part granularity perspective. Each resolution perspective is defined in reference to a combination of a proper countable parthood relation with a is-representationof relation and the defining properties of all subtypes of 'object aggregate' or 'object aggregate with fiat object part' that belong to the same compositional object aggregate of object granularity perspective or the same spatial fiat object part of object granularity perspective respectively (depicted is the
resolution perspective of cell aggregate with ECM, a particular resolution of object aggregate with fiat object part granularity perspective). Each resolution
granularity perspective contains only two levels: a lower level, to which subtypes of 'object cluster' or 'object cluster with portion of substance' (e.g.,
'cell cluster with portion of ECM') belong that represent countable representations of the corresponding subtypes of 'object aggregate' or 'object aggregate with fiat object part' (e.g., 'cell aggregate with ECM') respectively, and a higher level, to which subtypes of 'portion of substance' or 'portion
of granular mixture of substance X and Y' (e.g., 'portion of tissue') belong that represent the non-countable representations of the same object aggregate or object aggregate with fiat object part. ECM: extracellular matrix.
is based on a countable/non-countable representation-of
relation of an object aggregate or of an object aggregate
with fiat object part: (i) the countable 'object cluster' or
'object cluster with portion of substance' representation
and (ii) the non-countable 'portion of substance' or 'portion of granular mixture of substance X and Y' representation respectively (for an example see Fig. 10). The
countable/non-countable representation-of relation represents a subtype of the proper-part-of relation [32]. The
granulation criterion of a particular resolution perspective thus combines a proper parthood relation with a isrepresentation-of relation and the defining properties of
all subtypes of 'object aggregate' or 'object aggregate with
fiat object part' that belong to the same compositional
object of object aggregate granularity perspective or the
same basic spatial fiat object part of object granularity
perspective respectively:
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i) 'object cluster with portion of substance' iscountable-proper-part-of 'portion of granular mixture of
substance X and Y';
'portion of granular mixture of substance X and
Y' has-countable-proper-part 'object cluster with
portion of substance'.
And of limited relevance:
ii) 'object
cluster' is-countable-proper-partof 'portion of substance';
'portion
of
substance' has-countable-properpart 'object cluster'.
Particular resolution perspectives are demarcated from
one another according to the defining properties of the
respective subtypes of 'object aggregate' or 'object aggregate with fiat object part' that must belong to the same
compositional object aggregate of object granularity perspective or the same spatial fiat object part of object granularity perspective. In other words, each particular
resolution perspective is assigned to a particular compositional or spatial granularity perspective.
Each particular resolution perspective has granulation
of the scale dependent grain-size-according-to-resolution
granularity type (sgrG [32]). It uses a specific form of
proper parthood relation as its granulation relation and is
in this respect compatible to the other spatio-structural
granularity perspectives proposed above.
Each particular resolution perspective consists of only
two granularity levels. They are defined according to the
is-a-representation-of property and the defining properties of its corresponding subtype of 'object aggregate' or
'object aggregate with fiat object part'. They are demarcated from one another according to the property of
being countable or non-countable respectively. The granularity level to which the countable representations
belong represents the lower level, and entities belonging
to this lower level are related to entities belonging to the
higher level through is-countable-proper-part-of relations.
6.3 Integrating Different Spatio-Structural Granularity
Perspectives within a General Spatio-Structural Granularity
Framework
The different types of granularity perspectives proposed
above all meet the requirements put forward by Keet's
general theory of granularity [32]:
• All instances or types in any of the proposed granularity levels have at least one aspect in common,
which is the granulation criterion.
• Each proposed granularity perspective contains at
least two granularity levels.
• Adjacent fine and coarser-grained granularity levels
of any here proposed granularity perspective are
related to one another in two ways: (i) Instances or
types belonging to adjacent levels are related to one
Page 26 of 32
another according to the granulation relation. (ii) The
adjacent levels themselves are related to one another
according to a binary proper parthood relation RL
[32]. RL is identical for all levels and all granularity
perspectives [32]. This clear distinction between
relating levels on the one hand and relating their contents on the other hand is necessary since the latter,
the granulation relation, requires a more precise
specification than RL [32].
• Each level is contained in exactly one particular
granularity perspective. However, any particular
instance or type may belong to more than one perspective, but in each perspective it must belong only
to one granularity level. In the here proposed granularity perspectives it especially applies to objects,
which can belong to many different perspectives at
the same time.
The question is how the different structural granularity
perspectives can be combined within a common general
spatio-structural granularity framework. How do the different perspectives relate to each other, how do the levels
from different perspectives? How do the perspectives
relate to their overarching spatio-structural granularity
framework?
The integration of different granularity perspectives
within a common granularity framework can be achieved
by relating levels of distinct perspectives through relating
their perspectives [32]. The binary relation RP between
two distinct perspectives is irreflexive and symmetric
[32]. On the basis of this RP relation, according to Keet
[32], one can identify two strategies for directly linking
levels of different perspectives: (i) using overcrossing levels with mereology and (ii) using chaining levels with RL
and RP.
Two levels of different perspectives overcross if they,
although representing different levels of distinct perspectives, share at least some of their contents, which thus
overlap. And if levels of distinct perspectives overcross,
the perspectives themselves overcross as well [32]. This is
the case for all the granularity perspectives here proposed. They all share content in at least one of their levels
with another perspective. Together, they form a continuous chain or network of overcrossing perspectives - one
can traverse all here proposed perspectives through overlapping contents. All compositional perspectives discussed above, for instance, overcross with the
compositional object perspective or the object aggregate of
object aggregate perspective, of which the latter is of limited relevance for cumulative-constitutively organized
entities (see Fig. 6, 7, and 8). The two basic spatial granularity perspectives discussed above each share, for example, content of their higher granularity level with content
from one level of the compositional object granularity
perspective or the less relevant compositional object
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aggregate of object aggregate granularity perspective (see
Fig. 7, 8, and 9). The basic spatial granularity perspectives
as well as the additional compositional perspectives thus
all overcross with the compositional object granularity
perspective.
The scale dependent resolution of object aggregate with
fiat object part perspective and the less relevant resolution
of object aggregate perspective, on the other hand, do not
overcross with the compositional object granularity perspective. However, both of their granularity levels share
content with other spatial and compositional granularity
perspectives that contain subtypes of 'fiat object part' or
'object aggregate' in one of their granularity levels. Therefore they overcross with various additional compositional
and basic spatial granularity perspectives (for examples
see Fig. 6, 7, 8, and 9).
Within the here proposed general integrated spatiostructural granularity framework, the compositional
object granularity perspective takes in a central role, since
its levels provide the content that is shared with many of
the framework's other granularity perspectives. It thus
provides the backbone to the entire spatio-structural
granularity framework.
Now, all that is still required in order to integrate granularity levels, granularity perspectives, and the subject
domain into a common granularity framework, is another
relation which relates granularity levels with their perspective and perspectives with their subject domain.
According to Keet [32], this is a transitive and acyclic (i.e.
an object X does not have a path to itself ) binary containment relation RE, which is a subtype of the proper parthood relation (for an overview of the different relations
between the various components of Keet's general theory
of granularity see Figure 3.1, p. 48 in [32]).
6.4 Assigning Structural Granularity Values to SpatioStructural Granularity Levels
An interesting question that arises when looking at the
general integrated spatio-structural granularity framework presented here is whether it is possible to order all
granularity levels across different perspectives according
to an overarching single general spatio-structural granularity perspective. Such a perspective would have to be
very general and thus rather abstract in order to cover all
the different perspectives by overcrossing with all their
levels, which seems to be impossible on the type level.
However, when considering the structural organization of
a particular cumulative-constitutively organized material
entity, it is possible to assign a structural granularity
value to each level of granularity of each perspective of
the framework. In order to be able to do so, one has to
utilize the overcross-relations between different perspectives within the framework. Overcrossing levels must
share the same structural granularity value. Here, again,
Page 27 of 32
does the compositional object granularity perspective
function as a backbone. One can assign a fix natural number to each of its levels. The lowest level receives the
number one. Each subsequent level receives the next
higher natural number, so that adjacent fine and coarsergrained levels possess the structural granularity value X
and X+1, with X being an element of the natural numbers
(see Fig. 5). Now, all granularity levels of other granularity
perspectives that overcross the compositional object granularity perspective share the same fix structural granularity value with the corresponding overcrossing level. This
applies to the finer grained level of most of the additional
compositional perspectives and the higher level of the
basic spatial fiat object part of object granularity perspective (see Fig. 6 and 9).
Due to the arbitrariness of regional parts, however, the
non-overcrossing levels of both the additional compositional granularity perspectives and the basic spatial granularity perspectives cannot possess fix structural
granularity values. Instead, only relative structural granularity values can be assigned. These relative values are
assigned in relation to the fix values of the overcrossing
levels. The relative values are in between the fix values
and do not represent natural numbers anymore. For
instance the coarser-grained 'cell aggregate with ECM'
level of the compositional cell of cell aggregate with ECM
perspective possesses a structural granularity value that is
higher than the value of the finer grained 'cell' level, but
lower than the adjacent coarser-grained corresponding
'organ' level of the backbone compositional object granularity perspective (Fig. 6 and 7). Moreover, the coarsergrained 'cell aggregate with ECM' level of the compositional cell of cell aggregate with ECM perspective and the
finer-grained 'cell aggregate with ECM' level of the basic
spatial cell aggregate with ECM of organ perspective share
the same relative structural granularity value since their
contents overlap (Fig. 7). But then, the relative structural
granularity values of the two levels of for example the resolution of cell aggregate with ECM granularity perspective
both fall within the range of the relative structural granularity value of the corresponding 'cell aggregate with
ECM' level of both the spatial cell aggregate with ECM of
organ perspective and the compositional cell of cell aggregate with ECM perspective. Within this value range, however, the fine-grained 'cell cluster with ECM'
representation can receive a relative structural granularity value that is equal to or less than the value of the corresponding coarser-grained 'portion of tissue'
representation (Fig. 7 and 10) (see also discussion in the
next paragraph).
The resulting order of structural granularity values with
fix natural numbers linked to different types of objects on
the one hand and relative values linked to different types
of object aggregates and fiat object parts on the other
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hand seems to reflect an idea it has in common with the
Foundational Model of Anatomy (FMA [11]). The FMA
distinguishes salient levels of structural organization (i.e.
Biological macromolecule <Cell <Portion of tissue <Organ
<Organ system <Cardinal body part <Body), which are
defined in reference to units of structural organization
(i.e. objects and object aggregates) from transitional or
intermediate levels (i.e. Cardinal cell part <Cardinal tissue part <Cardinal organ part <Organ system subdivision
<Subdivision of cardinal body part), which are defined in
reference to subdivisions or cardinal parts of objects (i.e.
regional parts, fiat object parts). The transitive or intermediate nature that Rosse and Mejino [11] assert for
granularity levels containing regional parts is reflected in
the relative structural granularity values in the granularity
framework proposed here. However, the FMA framework
does not differentiate the different types of granularity
perspectives involved and thus suffers from the problems
discussed further above.
6.5 Additional Spatial Granularity Perspectives
One can introduce another type of spatial granularity
perspective in addition to those proposed above, a spatial fiat object part of fiat object part granularity perspective. Each fiat object part entity can itself be
partitioned into its regional parts, which at their turn can
be partitioned again. These partitions are based on
proper parthood relations between fiat object parts and
form what could be called second (or even higher) order
regional parts: they are regional parts of regional parts.
The granulation criterion of this additional type of spatial granularity perspective combines the proper parthood property with the defining properties of the
involved subtypes of 'fiat object part':
'fiat object part' is-proper-part-of
'fiat
object part';
'fiat object part' has-direct-proper-part 'fiat
object part'.
This additional spatial granularity perspective can also
be integrated into the general spatio-structural granularity framework: The additional spatial perspective overcrosses with other spatial and compositional perspectives
of the framework. The additional perspective concerns
only fiat object parts and not objects. As a consequence,
the respective levels do not overcross with any level of the
backbone compositional object granularity perspective,
each of which possesses a fix natural number structural
granularity value. Instead, they overcross with levels of
other perspectives, which at their turn only possess relative structural granularity values. Since these values represent real number intervals, they can accommodate an
infinite number of smaller real number intervals for
structural granularity values. This allows to assign struc-
Page 28 of 32
tural granularity value intervals to each level of the spatial fiat object part of fiat object part granularity
perspective that is within the range of relative structural
granularity values of the granularity levels with which the
perspective overcrosses (Fig. 11). The only thing that is
required is an additional finer-grained/coarser-grained
relation that further confines the structural granularity
value interval for the granularity levels of the spatial fiat
object part of fiat object part perspectives and that demarcates all granularity levels from one another that belong
to the same broader structural granularity value interval.
The problem with the additional spatial granularity
perspective is that a given object entity can be partitioned
in many different ways into various different regional
parts, some of which will even overlap one another. These
regional parts themselves could be partitioned into various different regional parts as well. While all these different regional parts necessarily represent proper parts of
their corresponding object entity and thus necessarily
belong to some of the above mentioned spatial granularity perspectives, they are not necessarily proper parts of
one another. For instance, not all possible fiat object parts
of a given object necessarily belong to the same granular
partition - some fiat object parts will overlap with one
another and only some will be proper parts of one
another. As a consequence, theoretically for any given
object there are an infinite number of different additional
spatial fiat object part of fiat object part granularity perspectives possible that contain 'fiat object part' levels,
although only one of them could be realized when having
to cut the object into actual fiat object parts. Therefore,
the application of spatial fiat object part of fiat object part
granularity perspectives should be restricted to instances
(i.e. particulars) and not to types (i.e. universals).
6.6 Scale Dependent Size Granularity Perspectives
One can also granulate all types of material entities solely
according to a specific type of scale. Such a quantitative
size granularity perspective would not rest on any type
of parthood relation. Instead, it is based on some largerthan/smaller-than relation in combination with a particular type of scale (i.e. all types of properties that are projectible on a metrical measure, e.g. max. width, surface,
volume, weight, max. lifespan). Arbitrary measurethresholds demarcate and order its various granularity
levels. Such granularity perspectives are usually only
applicable to the instance level - at least in biology, the
variability and diversity among instances of the same type
does not allow such a perspective to be applied to the
type level. Instead of structural granularity values,
respective granularity levels would have different scale
granularity values assigned.
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'Fiat
organ part 1'
'Cell'
Granular
Partition
Page 29 of 32
proper
part of
Across different
Perspectives
'Fiat
organ part 2'
proper
part of
'Fiat
organ part 3'
proper
part of
'Organ'
proper
part of
GB= X
GB< GC1 < GC2
GC1 < GC2 < GC3
GC2 < GC3 < GC
GC = X + 1
GB: Gran. Value of
this Level
GC1 : Gran. Value of
this Level
GC2 : Gran. Value of
this Level
GC3 : Gran. Value of
this Level
GC : Gran. Value of
this Level
Overcrosses
with e.g.:
Overcrosses
with e.g.:
Overcrosses
with e.g.:
Overcrosses
with e.g.:
qualitative
qualitative
qualitative
qualitative
Compositional
Spatial
Spatial
Spatial
Cell of Fiat
Organ Part
Fiat Organ Part
of Organ
Fiat Organ Part of
Fiat Organ Part
Fiat Organ Part
of Organ
Granularity
Granularity
Granularity
Granularity
Overcrosses
with e.g.:
qualitative
Compositional
Object
Granularity
Figure 11 Example of how Structural Granularity Values are assigned to a given Partition. Example for a particular granular partition of an organ
into its fiat organ part and cell parts. Shown is a chain of proper parthood relations, starting from a particular cell, that is part of a specific fiat organ
part 1, which at its turn is part of a fiat organ part 2 that is part of a fiat organ part 3, which is part of an organ. The structural granularity values of the
respective object entities (i.e. cell and organ) can be inferred using the fix natural number values from the compositional object granularity perspective: value X for the cell entity and value (X+1) for the organ entity. The structural granularity values of the different fiat organ part entities must be
within the range of X and (X+1), as the overcrossing spatial granularity perspectives imply. With the finer-grained relation ' < ', the respective value
interval 'X to (X+1)' can be further confined for each particular fiat organ part entity in the way it is shown in this figure.
7 Conclusions
The most important consequence from accepting the
existence of cumulative-constitutively organized material entities is that all their parts of one granularity level
do not always exhaustively sum to the whole in a granularity tree - not all entities belonging to one level of granularity always form parts of entities of the next higher
level of granularity (contradicting [36]). This has far
reaching consequences for any theory of granularity,
which have been discussed in detail above. Another problem of most commonly used granularity schemes is that
they commit to a granularity framework that does not
allow the integration of different granularity perspectives
within a common framework.
The problem that Kumar et al.'s [13] granularity
scheme for human anatomy had to face regarding their
basic granularity principles, with its levels of 'cardinal
body part' and 'organ system' violating their second (i.e.
grains in a given level are parts of grains in the next higher
level) and their fifth principle (i.e. grains in a given level
must be smaller in size than those entities on the next
higher level of which they are parts), can be avoided. The
problem does not occur when following the principles
mentioned in the previous section, and when precisely
distinguishing different granularity perspectives and integrating them to an overall spatio-structural granularity
framework, instead of forcing them into a single perspective. Transferred to the general integrated spatio-structural granularity framework presented here, a human
respiratory system would be an organ cluster with por-
tion of tissue and, due to its cumulative constitutive organization, a fiat body part. The human head and chest, on
the other hand, are fiat body parts as well. In their granularity scheme for human anatomy, however, Kumar et al.
[13] treat the latter (i.e. 'Cardinal body part') as the finer
grained level of granularity and the former (i.e. 'Organ
system') as the adjacent coarser grained level, although
both are actually fiat body parts. Transferred to the here
presented integrated framework, their structural granularity values occupy the same value range.
The only granularity perspective, in which both a cardinal body part and an organ system could belong to adjacent levels of granularity is a spatial fiat object part of fiat
object part granularity perspective. This perspective,
however, can be only applied reasonably to the granulation of individual material entities - i.e. to instances, but
not to types.
However, since both the respiratory system, as well as
the head and the chest, are proper parts of a human body,
and since both belong to granularity levels that share the
same structural granularity value range, there might be
some organisms in which some cardinal body part is
proper part of some organ system (and vice versa). But
this is not necessarily the case for every cardinal body
part and every organ system, and most certainly not for
every organism. Therefore, the respective types cannot
belong to adjacent granularity levels of a common type
granularity perspective.
With the here presented integrated granularity framework, any particular spatio-structural granular partition
Vogt BMC Bioinformatics 2010, 11:289
http://www.biomedcentral.com/1471-2105/11/289
of any given material entity can be modeled and to each
of its particular regional and constitutive parts an unambiguous structural granularity value can be assigned,
which consistently increases from each part to whole
relation. In other words, any particular proper parthood
relation between two material entities can be assigned to
one of the here proposed granularity perspectives. As a
consequence, the rather artificial distinction between
gross and granular parthood suggested by Rector et al.
[69], which differentiates between a gross parthood for
parthood relations between entities of the same granular
level, and a granular parthood for parthood relations
between entities belonging to different levels of granularity, is not required in the here presented granularity
framework.
Moreover, the framework does not require an a priori
ordering of parts - all that is required are the proper parthood relations. The assignment of the structural granularity values can be inferred on the basis of the
granularity framework, using the compositional object
perspective as the backbone that provides the fix natural
number values. In a next step, the various other compositional, spatial, and resolution-dependent perspectives are
accordingly joined with the compositional object perspective and with one another via overcrossing granularity
levels, contributing the relative real number structural
granularity intervals. This is possible because in the integrated framework, (i) all granularity perspectives are connected to one another through overcrossing granularity
levels, together forming an integrated whole with no perspective being cut off from the other perspectives; (ii)
every granularity perspective is based on a proper parthood relation. As a consequence, any spatio-structural
granularity partition can be modeled. Any given spatiostructural granular partition with already assigned structural granularity values can be granulated even further
(e.g., through adding levels to the spatial fiat object part
of fiat object part granularity perspective), without causing inconsistencies within the granularity framework and
without having to change any of the already assigned
structural granularity values.
The here presented framework provides a spatio-structural granularity framework not only for biological material entities, but for all domain reference ontologies that
model cumulative-constitutively organized material entities. With its multi-perspectives approach and the possibility to include even granularity perspectives that are
based on taxonomic inclusion (i.e. is-a relations), it allows
querying an ontology stored in a database at one's desired
different levels of detail: The contents of a database can
be organized according to diverse granularity perspectives, which at their turn provide different views on its
content (i.e. data, knowledge), each organized into different levels of detail. This not only allows for detailed and
Page 30 of 32
sophisticated searches within the database, enhancement
of its information and knowledge management, granular
zooming in and out, and abstracting away of irrelevant
details while focusing on the level of detail relevant for
one's specific information needs [32]. It also can be very
valuable for making inferences utilizing computer reasoning (e.g. [70,71]). It would even allow the use of different taxonomies. For example a taxonomy of
morphological types exclusively based on spatio-structural properties and organized through is-a relations,
alongside a taxonomy of functional types exclusively
based on dispositions and organized through is-functionally-a relations. Each taxonomy could follow the single
inheritance rule, and all relevant relations of taxonomic
inclusion still could be modeled in the same ontology
through different taxonomic granularity perspectives.
Granularity, in general, is best understood - at least
when used in combination with ontologies and databases
- as an additional layer on top of an existing domain reference or a terminology-based application ontology. It
functions like a meta-ontology, which organizes the
respective ontology according to some general granularity framework, thereby organizing the properties and
relations present in the underlying ontology according to
different granularity perspectives. Whereas the here presented general integrated spatio-structural granularity
framework is restricted to granularity perspectives that
are based on relative location (i.e. spatial partitions) and
structural composition (compositional partitions), both
of which rest on structural proper parthood relations, it
does not necessarily have to adhere to these restrictions.
It could, for instance, be integrated with a granularity
framework that is based on causal dispositions (i.e. functional parthood relations), on temporal partitions, on
developmental properties and relations, on genealogical
relations, or on evolutionary origin. This would facilitate
the often called for integration of life-science data, which
demands a single framework in which all different disciplines and their respective data types can be involved (see
also [12,13]).
Authors' contributions
LV developed the here presented framework and wrote the manuscript.
Acknowledgements
I thank Peter Grobe, Björn Quast, and Thomas Bartolomaeus for reading and
criticizing earlier drafts of this manuscript. I thank Alex and Nico for proofreading the manuscript for grammar and spelling. I also thank two anonymous
reviewers for giving valuable comments and suggestions. It goes without saying, however, that I am solely responsible for all the arguments and statements
in this paper. This study was supported by the German Research Foundation
DFG (VO 1244/3-2 & VO 1244/3-3). I am also grateful to the taxpayers of Germany.
Author Details
Institut für Evolutionsbiologie und Ökologie, Universität Bonn; An der
Immenburg 1, D-53121 Bonn, Germany
Vogt BMC Bioinformatics 2010, 11:289
http://www.biomedcentral.com/1471-2105/11/289
Received: 12 January 2010 Accepted: 28 May 2010
Published: 28 May 2010
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doi: 10.1186/1471-2105-11-289
Cite this article as: Vogt, Spatio-structural granularity of biological material
entities BMC Bioinformatics 2010, 11:289
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